[{"data":1,"prerenderedAt":11086},["ShallowReactive",2],{"post-\u002Fblog\u002F2023\u002F2023-12-07-how-do-computer-calculate-the-log-function":3},{"id":4,"title":5,"body":6,"description":12,"draft":11072,"enableComment":11073,"extension":11074,"image":11075,"meta":11076,"navigation":11073,"onday":11077,"path":11078,"seo":11079,"stem":11080,"summary":11081,"tags":11082,"__hash__":11085},"blog\u002Fblog\u002F2023\u002F2023-12-07-How-do-computer-calculate-the-log-function.md","How Do Computer Calculate the Log-Function",{"type":7,"value":8,"toc":11065},"minimark",[9,13,16,331,415,458,461,1244,1363,1760,1794,2626,2869,2872,3988,4030,4309,5345,5348,6522,6525,7819,7822,8305,8644,8676,9353,9913,10003,10006,10141,10218,10493,10757,10760,11020,11062],[10,11,12],"p",{},"Brief Introduction to the Implementation Approach in glibc:",[10,14,15],{},"For the logarithm of any base, we can utilize the change of base 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term ",[17,335,337,355],{"className":336,"translate":21},[25],[17,338,340],{"className":339},[29],[31,341,342],{"xmlns":33},[36,343,344,352],{},[52,345,346,348,350],{},[60,347,83],{},[64,349,66],{},[60,351,69],{},[105,353,354],{"encoding":107},"\\ln k",[17,356,358],{"className":357,"ariaHidden":113},[112],[17,359,361,365,368,371],{"className":360},[117],[17,362],{"className":363,"style":364},[121],"height:0.6944em;",[17,366,83],{"className":367},[126],[17,369],{"className":370,"style":192},[191],[17,372,69],{"className":373,"style":173},[171,172]," in equation (1) is a constant, so for logarithms of any base, we only need to calculate ",[17,376,378,396],{"className":377,"translate":21},[25],[17,379,381],{"className":380},[29],[31,382,383],{"xmlns":33},[36,384,385,393],{},[52,386,387,389,391],{},[60,388,83],{},[64,390,66],{},[60,392,72],{},[105,394,395],{"encoding":107},"\\ln x",[17,397,399],{"className":398,"ariaHidden":113},[112],[17,400,402,405,408,411],{"className":401},[117],[17,403],{"className":404,"style":364},[121],[17,406,83],{"className":407},[126],[17,409],{"className":410,"style":192},[191],[17,412,72],{"className":413},[171,172],".",[416,417,419,420],"h3",{"id":418},"computation-of-lnxln-xlnx","Computation of ",[17,421,423,440],{"className":422,"translate":21},[25],[17,424,426],{"className":425},[29],[31,427,428],{"xmlns":33},[36,429,430,438],{},[52,431,432,434,436],{},[60,433,83],{},[64,435,66],{},[60,437,72],{},[105,439,395],{"encoding":107},[17,441,443],{"className":442,"ariaHidden":113},[112],[17,444,446,449,452,455],{"className":445},[117],[17,447],{"className":448,"style":364},[121],[17,450,83],{"className":451},[126],[17,453],{"className":454,"style":192},[191],[17,456,72],{"className":457},[171,172],[10,459,460],{},"In advanced calculus, we learned the following Taylor series:",[17,462,464],{"className":463,"translate":21},[20],[17,465,467,613],{"className":466,"translate":21},[25],[17,468,470],{"className":469},[29],[31,471,472],{"xmlns":33,"display":34},[36,473,474,610],{},[39,475,476],{"width":41},[43,477,478,480,603,605],{},[46,479],{"width":48},[46,481,482],{},[52,483,484,486,488,491,494,497,499,501,503,505,508,515,522,524,531,537,539,546,552,554,557,559,577],{},[60,485,83],{},[64,487,66],{},[64,489,319],{"stretchy":490},"false",[492,493,326],"mn",{},[64,495,496],{},"+",[60,498,72],{},[64,500,330],{"stretchy":490},[64,502,75],{},[60,504,72],{},[64,506,507],{},"−",[77,509,510,512],{},[492,511,326],{},[492,513,514],{},"2",[516,517,518,520],"msup",{},[60,519,72],{},[492,521,514],{},[64,523,496],{},[77,525,526,528],{},[492,527,326],{},[492,529,530],{},"3",[516,532,533,535],{},[60,534,72],{},[492,536,530],{},[64,538,507],{},[77,540,541,543],{},[492,542,326],{},[492,544,545],{},"4",[516,547,548,550],{},[60,549,72],{},[492,551,545],{},[64,553,496],{},[64,555,556],{},"⋯",[64,558,75],{},[560,561,562,565,573],"munderover",{},[64,563,564],{},"∑",[52,566,567,569,571],{},[60,568,69],{},[64,570,75],{},[492,572,326],{},[60,574,576],{"mathvariant":575},"normal","∞",[77,578,579,601],{},[52,580,581,583,585,587,599],{},[64,582,319],{"stretchy":490},[64,584,507],{},[492,586,326],{},[516,588,589,591],{},[64,590,330],{"stretchy":490},[52,592,593,595,597],{},[60,594,69],{},[64,596,507],{},[492,598,326],{},[60,600,72],{},[60,602,69],{},[46,604],{"width":48},[46,606,607],{},[101,608,609],{},"(2)",[105,611,612],{"encoding":107},"\\ln(1+x) = x - \\frac{1}{2}x^2 + \\frac{1}{3}x^3 - \\frac{1}{4}x^4 + \\dots = \\sum_{k=1}^{\\infty}\\frac{(-1)^{k-1}x}{k} 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we want to find ",[17,1248,1250,1267],{"className":1249,"translate":21},[25],[17,1251,1253],{"className":1252},[29],[31,1254,1255],{"xmlns":33},[36,1256,1257,1265],{},[52,1258,1259,1261,1263],{},[60,1260,83],{},[64,1262,66],{},[60,1264,72],{},[105,1266,395],{"encoding":107},[17,1268,1270],{"className":1269,"ariaHidden":113},[112],[17,1271,1273,1276,1279,1282],{"className":1272},[117],[17,1274],{"className":1275,"style":364},[121],[17,1277,83],{"className":1278},[126],[17,1280],{"className":1281,"style":192},[191],[17,1283,72],{"className":1284},[171,172],"; naturally, we can set ",[17,1287,1289,1312],{"className":1288,"translate":21},[25],[17,1290,1292],{"className":1291},[29],[31,1293,1294],{"xmlns":33},[36,1295,1296,1309],{},[52,1297,1298,1301,1303,1305,1307],{},[60,1299,1300],{},"t",[64,1302,75],{},[60,1304,72],{},[64,1306,507],{},[492,1308,326],{},[105,1310,1311],{"encoding":107},"t=x-1",[17,1313,1315,1334,1352],{"className":1314,"ariaHidden":113},[112],[17,1316,1318,1322,1325,1328,1331],{"className":1317},[117],[17,1319],{"className":1320,"style":1321},[121],"height:0.6151em;",[17,1323,1300],{"className":1324},[171,172],[17,1326],{"className":1327,"style":202},[191],[17,1329,75],{"className":1330},[206],[17,1332],{"className":1333,"style":202},[191],[17,1335,1337,1340,1343,1346,1349],{"className":1336},[117],[17,1338],{"className":1339,"style":670},[121],[17,1341,72],{"className":1342},[171,172],[17,1344],{"className":1345,"style":635},[191],[17,1347,507],{"className":1348},[639],[17,1350],{"className":1351,"style":635},[191],[17,1353,1355,1359],{"className":1354},[117],[17,1356],{"className":1357,"style":1358},[121],"height:0.6444em;",[17,1360,326],{"className":1361},[171],", then:",[17,1364,1366],{"className":1365,"translate":21},[20],[17,1367,1369,1461],{"className":1368,"translate":21},[25],[17,1370,1372],{"className":1371},[29],[31,1373,1374],{"xmlns":33,"display":34},[36,1375,1376,1458],{},[39,1377,1378],{"width":41},[43,1379,1380,1382,1452,1454],{},[46,1381],{"width":48},[46,1383,1384],{},[52,1385,1386,1388,1390,1392,1394,1396,1398,1400,1402,1404,1406,1408,1410,1412,1414,1428,1446],{},[60,1387,83],{},[64,1389,66],{},[64,1391,319],{"stretchy":490},[60,1393,72],{},[64,1395,330],{"stretchy":490},[64,1397,75],{},[60,1399,83],{},[64,1401,66],{},[64,1403,319],{"stretchy":490},[492,1405,326],{},[64,1407,496],{},[60,1409,1300],{},[64,1411,330],{"stretchy":490},[64,1413,75],{},[560,1415,1416,1418,1426],{},[64,1417,564],{},[52,1419,1420,1422,1424],{},[60,1421,69],{},[64,1423,75],{},[492,1425,326],{},[60,1427,576],{"mathvariant":575},[77,1429,1430,1444],{},[52,1431,1432,1434,1436,1438],{},[64,1433,319],{"stretchy":490},[64,1435,507],{},[492,1437,326],{},[516,1439,1440,1442],{},[64,1441,330],{"stretchy":490},[60,1443,69],{},[60,1445,69],{},[516,1447,1448,1450],{},[60,1449,1300],{},[60,1451,69],{},[46,1453],{"width":48},[46,1455,1456],{},[101,1457,609],{},[105,1459,1460],{"encoding":107},"\\ln(x) = \\ln(1+t) = \\sum_{k=1}^{\\infty}\\frac{(-1)^k}{k}t^k \\tag{2}",[17,1462,1464,1491,1515,1536,1739],{"className":1463,"ariaHidden":113},[112],[17,1465,1467,1470,1473,1476,1479,1482,1485,1488],{"className":1466},[117],[17,1468],{"className":1469,"style":622},[121],[17,1471,83],{"className":1472},[126],[17,1474,319],{"className":1475},[223],[17,1477,72],{"className":1478},[171,172],[17,1480,330],{"className":1481},[304],[17,1483],{"className":1484,"style":202},[191],[17,1486,75],{"className":1487},[206],[17,1489],{"className":1490,"style":202},[191],[17,1492,1494,1497,1500,1503,1506,1509,1512],{"className":1493},[117],[17,1495],{"className":1496,"style":622},[121],[17,1498,83],{"className":1499},[126],[17,1501,319],{"className":1502},[223],[17,1504,326],{"className":1505},[171],[17,1507],{"className":1508,"style":635},[191],[17,1510,496],{"className":1511},[639],[17,1513],{"className":1514,"style":635},[191],[17,1516,1518,1521,1524,1527,1530,1533],{"className":1517},[117],[17,1519],{"className":1520,"style":622},[121],[17,1522,1300],{"className":1523},[171,172],[17,1525,330],{"className":1526},[304],[17,1528],{"className":1529,"style":202},[191],[17,1531,75],{"className":1532},[206],[17,1534],{"className":1535,"style":202},[191],[17,1537,1539,1542,1609,1612,1709],{"className":1538},[117],[17,1540],{"className":1541,"style":1031},[121],[17,1543,1545],{"className":1544},[126,1035],[17,1546,1548,1601],{"className":1547},[142,143],[17,1549,1551,1598],{"className":1550},[147],[17,1552,1554,1574,1584],{"className":1553,"style":1045},[151],[17,1555,1556,1559],{"style":1048},[17,1557],{"className":1558,"style":1052},[159],[17,1560,1562],{"className":1561},[164,165,166,167],[17,1563,1565,1568,1571],{"className":1564},[171,167],[17,1566,69],{"className":1567,"style":173},[171,172,167],[17,1569,75],{"className":1570},[206,167],[17,1572,326],{"className":1573},[171,167],[17,1575,1576,1579],{"style":1070},[17,1577],{"className":1578,"style":1052},[159],[17,1580,1581],{},[17,1582,564],{"className":1583},[126,1079,1080],[17,1585,1586,1589],{"style":1083},[17,1587],{"className":1588,"style":1052},[159],[17,1590,1592],{"className":1591},[164,165,166,167],[17,1593,1595],{"className":1594},[171,167],[17,1596,576],{"className":1597},[171,167],[17,1599,178],{"className":1600},[177],[17,1602,1604],{"className":1603},[147],[17,1605,1607],{"className":1606,"style":1105},[151],[17,1608],{},[17,1610],{"className":1611,"style":192},[191],[17,1613,1615,1618,1706],{"className":1614},[171],[17,1616],{"className":1617},[223,224],[17,1619,1621],{"className":1620},[77],[17,1622,1624,1698],{"className":1623},[142,143],[17,1625,1627,1695],{"className":1626},[147],[17,1628,1630,1641,1649],{"className":1629,"style":1129},[151],[17,1631,1632,1635],{"style":240},[17,1633],{"className":1634,"style":244},[159],[17,1636,1638],{"className":1637},[171],[17,1639,69],{"className":1640,"style":173},[171,172],[17,1642,1643,1646],{"style":259},[17,1644],{"className":1645,"style":244},[159],[17,1647],{"className":1648,"style":267},[266],[17,1650,1651,1654],{"style":270},[17,1652],{"className":1653,"style":244},[159],[17,1655,1657,1660,1663,1666],{"className":1656},[171],[17,1658,319],{"className":1659},[223],[17,1661,507],{"className":1662},[171],[17,1664,326],{"className":1665},[171],[17,1667,1669,1672],{"className":1668},[304],[17,1670,330],{"className":1671},[304],[17,1673,1675],{"className":1674},[138],[17,1676,1678],{"className":1677},[142],[17,1679,1681],{"className":1680},[147],[17,1682,1684],{"className":1683,"style":1184},[151],[17,1685,1686,1689],{"style":1187},[17,1687],{"className":1688,"style":160},[159],[17,1690,1692],{"className":1691},[164,165,166,167],[17,1693,69],{"className":1694,"style":173},[171,172,167],[17,1696,178],{"className":1697},[177],[17,1699,1701],{"className":1700},[147],[17,1702,1704],{"className":1703,"style":298},[151],[17,1705],{},[17,1707],{"className":1708},[304,224],[17,1710,1712,1715],{"className":1711},[171],[17,1713,1300],{"className":1714},[171,172],[17,1716,1718],{"className":1717},[138],[17,1719,1721],{"className":1720},[142],[17,1722,1724],{"className":1723},[147],[17,1725,1728],{"className":1726,"style":1727},[151],"height:0.8991em;",[17,1729,1730,1733],{"style":774},[17,1731],{"className":1732,"style":160},[159],[17,1734,1736],{"className":1735},[164,165,166,167],[17,1737,69],{"className":1738,"style":173},[171,172,167],[17,1740,1742,1745],{"className":1741},[308],[17,1743],{"className":1744,"style":1031},[121],[17,1746,1748,1751,1757],{"className":1747},[171,315],[17,1749,319],{"className":1750},[171],[17,1752,1754],{"className":1753},[171],[17,1755,514],{"className":1756},[171],[17,1758,330],{"className":1759},[171],[10,1761,1762,1763,1793],{},"However, expanding the series to infinity in practical calculations is often not feasible. If we expand it to the ",[17,1764,1766,1780],{"className":1765,"translate":21},[25],[17,1767,1769],{"className":1768},[29],[31,1770,1771],{"xmlns":33},[36,1772,1773,1778],{},[52,1774,1775],{},[60,1776,1777],{},"n",[105,1779,1777],{"encoding":107},[17,1781,1783],{"className":1782,"ariaHidden":113},[112],[17,1784,1786,1790],{"className":1785},[117],[17,1787],{"className":1788,"style":1789},[121],"height:0.4306em;",[17,1791,1777],{"className":1792},[171,172],"th term, the Taylor series with the Peano remainder term can be expressed as:",[17,1795,1797],{"className":1796,"translate":21},[20],[17,1798,1800,1946],{"className":1799,"translate":21},[25],[17,1801,1803],{"className":1802},[29],[31,1804,1805],{"xmlns":33,"display":34},[36,1806,1807,1943],{},[39,1808,1809],{"width":41},[43,1810,1811,1813,1936,1938],{},[46,1812],{"width":48},[46,1814,1815],{},[52,1816,1817,1819,1821,1823,1825,1827,1829,1831,1833,1835,1837,1843,1849,1851,1857,1863,1865,1871,1877,1879,1881,1883,1897,1915,1921,1923,1926,1928,1934],{},[60,1818,83],{},[64,1820,66],{},[64,1822,319],{"stretchy":490},[492,1824,326],{},[64,1826,496],{},[60,1828,1300],{},[64,1830,330],{"stretchy":490},[64,1832,75],{},[60,1834,1300],{},[64,1836,507],{},[77,1838,1839,1841],{},[492,1840,326],{},[492,1842,514],{},[516,1844,1845,1847],{},[60,1846,1300],{},[492,1848,514],{},[64,1850,496],{},[77,1852,1853,1855],{},[492,1854,326],{},[492,1856,530],{},[516,1858,1859,1861],{},[60,1860,1300],{},[492,1862,530],{},[64,1864,507],{},[77,1866,1867,1869],{},[492,1868,326],{},[492,1870,545],{},[516,1872,1873,1875],{},[60,1874,1300],{},[492,1876,545],{},[64,1878,496],{},[64,1880,556],{},[64,1882,75],{},[560,1884,1885,1887,1895],{},[64,1886,564],{},[52,1888,1889,1891,1893],{},[60,1890,69],{},[64,1892,75],{},[492,1894,326],{},[60,1896,1777],{},[77,1898,1899,1913],{},[52,1900,1901,1903,1905,1907],{},[64,1902,319],{"stretchy":490},[64,1904,507],{},[492,1906,326],{},[516,1908,1909,1911],{},[64,1910,330],{"stretchy":490},[60,1912,69],{},[60,1914,69],{},[516,1916,1917,1919],{},[60,1918,1300],{},[60,1920,69],{},[64,1922,496],{},[60,1924,1925],{},"o",[64,1927,319],{"stretchy":490},[516,1929,1930,1932],{},[60,1931,1300],{},[60,1933,1777],{},[64,1935,330],{"stretchy":490},[46,1937],{"width":48},[46,1939,1940],{},[101,1941,1942],{},"(3)",[105,1944,1945],{"encoding":107},"\\ln(1+t) = t - \\frac{1}{2}t^2 + \\frac{1}{3}t^3 - \\frac{1}{4}t^4 + \\dots = \\sum_{k=1}^{n}\\frac{(-1)^k}{k}t^k + o(t^n) 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594],{"className":2595,"style":160},[159],[17,2597,2599],{"className":2598},[164,165,166,167],[17,2600,1777],{"className":2601},[171,172,167],[17,2603,330],{"className":2604},[304],[17,2606,2608,2611],{"className":2607},[308],[17,2609],{"className":2610,"style":1031},[121],[17,2612,2614,2617,2623],{"className":2613},[171,315],[17,2615,319],{"className":2616},[171],[17,2618,2620],{"className":2619},[171],[17,2621,530],{"className":2622},[171],[17,2624,330],{"className":2625},[171],[10,2627,2628,2629,2704,2705,2758,2759,2811,2812,2840,2841,414],{},"Note that the Peano remainder term here is ",[17,2630,2632,2656],{"className":2631,"translate":21},[25],[17,2633,2635],{"className":2634},[29],[31,2636,2637],{"xmlns":33},[36,2638,2639,2653],{},[52,2640,2641,2643,2645,2651],{},[60,2642,1925],{},[64,2644,319],{"stretchy":490},[516,2646,2647,2649],{},[60,2648,1300],{},[60,2650,1777],{},[64,2652,330],{"stretchy":490},[105,2654,2655],{"encoding":107},"o(t^n)",[17,2657,2659],{"className":2658,"ariaHidden":113},[112],[17,2660,2662,2665,2668,2671,2701],{"className":2661},[117],[17,2663],{"className":2664,"style":622},[121],[17,2666,1925],{"className":2667},[171,172],[17,2669,319],{"className":2670},[223],[17,2672,2674,2677],{"className":2673},[171],[17,2675,1300],{"className":2676},[171,172],[17,2678,2680],{"className":2679},[138],[17,2681,2683],{"className":2682},[142],[17,2684,2686],{"className":2685},[147],[17,2687,2690],{"className":2688,"style":2689},[151],"height:0.6644em;",[17,2691,2692,2695],{"style":1187},[17,2693],{"className":2694,"style":160},[159],[17,2696,2698],{"className":2697},[164,165,166,167],[17,2699,1777],{"className":2700},[171,172,167],[17,2702,330],{"className":2703},[304],", representing the error term. If ",[17,2706,2708,2727],{"className":2707,"translate":21},[25],[17,2709,2711],{"className":2710},[29],[31,2712,2713],{"xmlns":33},[36,2714,2715,2724],{},[52,2716,2717,2719,2722],{},[60,2718,1300],{},[64,2720,2721],{},">",[492,2723,326],{},[105,2725,2726],{"encoding":107},"t>1",[17,2728,2730,2749],{"className":2729,"ariaHidden":113},[112],[17,2731,2733,2737,2740,2743,2746],{"className":2732},[117],[17,2734],{"className":2735,"style":2736},[121],"height:0.6542em;vertical-align:-0.0391em;",[17,2738,1300],{"className":2739},[171,172],[17,2741],{"className":2742,"style":202},[191],[17,2744,2721],{"className":2745},[206],[17,2747],{"className":2748,"style":202},[191],[17,2750,2752,2755],{"className":2751},[117],[17,2753],{"className":2754,"style":1358},[121],[17,2756,326],{"className":2757},[171]," (or ",[17,2760,2762,2780],{"className":2761,"translate":21},[25],[17,2763,2765],{"className":2764},[29],[31,2766,2767],{"xmlns":33},[36,2768,2769,2777],{},[52,2770,2771,2773,2775],{},[60,2772,72],{},[64,2774,2721],{},[492,2776,514],{},[105,2778,2779],{"encoding":107},"x > 2",[17,2781,2783,2802],{"className":2782,"ariaHidden":113},[112],[17,2784,2786,2790,2793,2796,2799],{"className":2785},[117],[17,2787],{"className":2788,"style":2789},[121],"height:0.5782em;vertical-align:-0.0391em;",[17,2791,72],{"className":2792},[171,172],[17,2794],{"className":2795,"style":202},[191],[17,2797,2721],{"className":2798},[206],[17,2800],{"className":2801,"style":202},[191],[17,2803,2805,2808],{"className":2804},[117],[17,2806],{"className":2807,"style":1358},[121],[17,2809,514],{"className":2810},[171],"), the error in the above formula may become significantly large with the increase of ",[17,2813,2815,2828],{"className":2814,"translate":21},[25],[17,2816,2818],{"className":2817},[29],[31,2819,2820],{"xmlns":33},[36,2821,2822,2826],{},[52,2823,2824],{},[60,2825,1300],{},[105,2827,1300],{"encoding":107},[17,2829,2831],{"className":2830,"ariaHidden":113},[112],[17,2832,2834,2837],{"className":2833},[117],[17,2835],{"className":2836,"style":1321},[121],[17,2838,1300],{"className":2839},[171,172]," or ",[17,2842,2844,2857],{"className":2843,"translate":21},[25],[17,2845,2847],{"className":2846},[29],[31,2848,2849],{"xmlns":33},[36,2850,2851,2855],{},[52,2852,2853],{},[60,2854,72],{},[105,2856,72],{"encoding":107},[17,2858,2860],{"className":2859,"ariaHidden":113},[112],[17,2861,2863,2866],{"className":2862},[117],[17,2864],{"className":2865,"style":1789},[121],[17,2867,72],{"className":2868},[171,172],[10,2870,2871],{},"However, another formula is available:",[17,2873,2875],{"className":2874,"translate":21},[20],[17,2876,2878,3089],{"className":2877,"translate":21},[25],[17,2879,2881],{"className":2880},[29],[31,2882,2883],{"xmlns":33,"display":34},[36,2884,2885,3086],{},[39,2886,2887],{"width":41},[43,2888,2889,2891,3079,3081],{},[46,2890],{"width":48},[46,2892,2893],{},[52,2894,2895,2897,2899,2901,2903,2905,3010,3012,3014,3029,3043],{},[60,2896,83],{},[64,2898,66],{},[60,2900,72],{},[64,2902,75],{},[492,2904,514],{},[52,2906,2907,2909,2927,2929,2935,2963,2965,2972,3000,3002,3005,3008],{},[64,2908,319],{"fence":113},[77,2910,2911,2919],{},[52,2912,2913,2915,2917],{},[60,2914,72],{},[64,2916,507],{},[492,2918,326],{},[52,2920,2921,2923,2925],{},[60,2922,72],{},[64,2924,496],{},[492,2926,326],{},[64,2928,496],{},[77,2930,2931,2933],{},[492,2932,326],{},[492,2934,530],{},[516,2936,2937,2961],{},[52,2938,2939,2941,2959],{},[64,2940,319],{"fence":113},[77,2942,2943,2951],{},[52,2944,2945,2947,2949],{},[60,2946,72],{},[64,2948,507],{},[492,2950,326],{},[52,2952,2953,2955,2957],{},[60,2954,72],{},[64,2956,496],{},[492,2958,326],{},[64,2960,330],{"fence":113},[492,2962,530],{},[64,2964,496],{},[77,2966,2967,2969],{},[492,2968,326],{},[492,2970,2971],{},"5",[516,2973,2974,2998],{},[52,2975,2976,2978,2996],{},[64,2977,319],{"fence":113},[77,2979,2980,2988],{},[52,2981,2982,2984,2986],{},[60,2983,72],{},[64,2985,507],{},[492,2987,326],{},[52,2989,2990,2992,2994],{},[60,2991,72],{},[64,2993,496],{},[492,2995,326],{},[64,2997,330],{"fence":113},[492,2999,2971],{},[64,3001,496],{},[64,3003,3004],{},"…",[101,3006,3007],{}," ",[64,3009,330],{"fence":113},[64,3011,75],{},[492,3013,514],{},[560,3015,3016,3018,3027],{},[64,3017,564],{},[52,3019,3020,3022,3024],{},[60,3021,69],{},[64,3023,75],{},[492,3025,3026],{},"0",[60,3028,576],{"mathvariant":575},[77,3030,3031,3033],{},[492,3032,326],{},[52,3034,3035,3037,3039,3041],{},[492,3036,514],{},[60,3038,69],{},[64,3040,496],{},[492,3042,326],{},[516,3044,3045,3069],{},[52,3046,3047,3049,3067],{},[64,3048,319],{"fence":113},[77,3050,3051,3059],{},[52,3052,3053,3055,3057],{},[60,3054,72],{},[64,3056,507],{},[492,3058,326],{},[52,3060,3061,3063,3065],{},[60,3062,72],{},[64,3064,496],{},[492,3066,326],{},[64,3068,330],{"fence":113},[52,3070,3071,3073,3075,3077],{},[492,3072,514],{},[60,3074,69],{},[64,3076,496],{},[492,3078,326],{},[46,3080],{"width":48},[46,3082,3083],{},[101,3084,3085],{},"(4)",[105,3087,3088],{"encoding":107},"\\ln x = 2\\left(\\frac{x-1}{x+1} + \\frac{1}{3}\\left(\\frac{x-1}{x+1}\\right)^3 + \\frac{1}{5}\\left(\\frac{x-1}{x+1}\\right)^5 + \\dots\\right) = 2\\sum_{k=0}^{\\infty}\\frac{1}{2k+1}\\left(\\frac{x-1}{x+1}\\right)^{2k+1} 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an informal proof for equation ",[17,3992,3994,4011],{"className":3993,"translate":21},[25],[17,3995,3997],{"className":3996},[29],[31,3998,3999],{"xmlns":33},[36,4000,4001,4009],{},[52,4002,4003,4005,4007],{},[64,4004,319],{"stretchy":490},[492,4006,545],{},[64,4008,330],{"stretchy":490},[105,4010,3085],{"encoding":107},[17,4012,4014],{"className":4013,"ariaHidden":113},[112],[17,4015,4017,4020,4023,4026],{"className":4016},[117],[17,4018],{"className":4019,"style":622},[121],[17,4021,319],{"className":4022},[223],[17,4024,545],{"className":4025},[171],[17,4027,330],{"className":4028},[304],":",[10,4031,4032,4033,4061,4062,4100,4101,4204,4205,4308],{},"Replace ",[17,4034,4036,4049],{"className":4035,"translate":21},[25],[17,4037,4039],{"className":4038},[29],[31,4040,4041],{"xmlns":33},[36,4042,4043,4047],{},[52,4044,4045],{},[60,4046,72],{},[105,4048,72],{"encoding":107},[17,4050,4052],{"className":4051,"ariaHidden":113},[112],[17,4053,4055,4058],{"className":4054},[117],[17,4056],{"className":4057,"style":1789},[121],[17,4059,72],{"className":4060},[171,172]," in equation ",[17,4063,4065,4082],{"className":4064,"translate":21},[25],[17,4066,4068],{"className":4067},[29],[31,4069,4070],{"xmlns":33},[36,4071,4072,4080],{},[52,4073,4074,4076,4078],{},[64,4075,319],{"stretchy":490},[492,4077,514],{},[64,4079,330],{"stretchy":490},[105,4081,609],{"encoding":107},[17,4083,4085],{"className":4084,"ariaHidden":113},[112],[17,4086,4088,4091,4094,4097],{"className":4087},[117],[17,4089],{"className":4090,"style":622},[121],[17,4092,319],{"className":4093},[223],[17,4095,514],{"className":4096},[171],[17,4098,330],{"className":4099},[304]," with ",[17,4102,4104,4122],{"className":4103,"translate":21},[25],[17,4105,4107],{"className":4106},[29],[31,4108,4109],{"xmlns":33},[36,4110,4111,4119],{},[52,4112,4113],{},[77,4114,4115,4117],{},[492,4116,326],{},[60,4118,72],{},[105,4120,4121],{"encoding":107},"\\frac{1}{x}",[17,4123,4125],{"className":4124,"ariaHidden":113},[112],[17,4126,4128,4132],{"className":4127},[117],[17,4129],{"className":4130,"style":4131},[121],"height:1.1901em;vertical-align:-0.345em;",[17,4133,4135,4138,4201],{"className":4134},[171],[17,4136],{"className":4137},[223,224],[17,4139,4141],{"className":4140},[77],[17,4142,4144,4192],{"className":4143},[142,143],[17,4145,4147,4189],{"className":4146},[147],[17,4148,4151,4166,4174],{"className":4149,"style":4150},[151],"height:0.8451em;",[17,4152,4154,4157],{"style":4153},"top:-2.655em;",[17,4155],{"className":4156,"style":244},[159],[17,4158,4160],{"className":4159},[164,165,166,167],[17,4161,4163],{"className":4162},[171,167],[17,4164,72],{"className":4165},[171,172,167],[17,4167,4168,4171],{"style":259},[17,4169],{"className":4170,"style":244},[159],[17,4172],{"className":4173,"style":267},[266],[17,4175,4177,4180],{"style":4176},"top:-3.394em;",[17,4178],{"className":4179,"style":244},[159],[17,4181,4183],{"className":4182},[164,165,166,167],[17,4184,4186],{"className":4185},[171,167],[17,4187,326],{"className":4188},[171,167],[17,4190,178],{"className":4191},[177],[17,4193,4195],{"className":4194},[147],[17,4196,4199],{"className":4197,"style":4198},[151],"height:0.345em;",[17,4200],{},[17,4202],{"className":4203},[304,224]," 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&= \\ln\\left(\\frac{x+1}{x}\\right) = \\sum_{k=1}^{\\infty}\\frac{(-1)^k}{k}\\frac{1}{x^k} \\\\\n\\ln\\left(1-\\frac{1}{x}\\right) &= 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&= -\\sum_{k=1}^{\\infty}\\frac{1}{kx^k} \\\\\n\\Longrightarrow -\\ln\\left(\\frac{x-1}{x}\\right) &= \\sum_{k=1}^{\\infty}\\frac{1}{kx^k} \\\\\n\\Longrightarrow \\ln\\left(\\frac{x}{x-1}\\right) &= 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= 2\\sum_{k=0}^n{\\frac{1}{2k+1}\\frac{1}{x^{2k+1}}}",[17,7921,7923,8046],{"className":7922,"ariaHidden":113},[112],[17,7924,7926,7930,7933,7936,8037,8040,8043],{"className":7925},[117],[17,7927],{"className":7928,"style":7929},[121],"height:2.4em;vertical-align:-0.95em;",[17,7931,83],{"className":7932},[126],[17,7934],{"className":7935,"style":192},[191],[17,7937,7939,7945,8031],{"className":7938},[1015],[17,7940,7942],{"className":7941,"style":3136},[223,3135],[17,7943,319],{"className":7944},[3140,166],[17,7946,7948,7951,8028],{"className":7947},[171],[17,7949],{"className":7950},[223,224],[17,7952,7954],{"className":7953},[77],[17,7955,7957,8020],{"className":7956},[142,143],[17,7958,7960,8017],{"className":7959},[147],[17,7961,7963,7986,7994],{"className":7962,"style":708},[151],[17,7964,7965,7968],{"style":240},[17,7966],{"className":7967,"style":244},[159],[17,7969,7971,7974,7977,7980,7983],{"className":7970},[171],[17,7972,72],{"className":7973},[171,172],[17,7975],{"className":7976,"style":635},[191],[17,7978,507],{"className":7979},[639],[17,7981],{"className":7982,"style":635},[191],[17,7984,326],{"className":7985},[171],[17,7987,7988,7991],{"style":259},[17,7989],{"className":7990,"style":244},[159],[17,7992],{"className":7993,"style":267},[266],[17,7995,7996,7999],{"style":270},[17,7997],{"className":7998,"style":244},[159],[17,8000,8002,8005,8008,8011,8014],{"className":8001},[171],[17,8003,72],{"className":8004},[171,172],[17,8006],{"className":8007,"style":635},[191],[17,8009,496],{"className":8010},[639],[17,8012],{"className":8013,"style":635},[191],[17,8015,326],{"className":8016},[171],[17,8018,178],{"className":8019},[177],[17,8021,8023],{"className":8022},[147],[17,8024,8026],{"className":8025,"style":3223},[151],[17,8027],{},[17,8029],{"className":8030},[304,224],[17,8032,8034],{"className":8033,"style":3136},[304,3135],[17,8035,330],{"className":8036},[3140,166],[17,8038],{"className":8039,"style":202},[191],[17,8041,75],{"className":8042},[206],[17,8044],{"className":8045,"style":202},[191],[17,8047,8049,8052,8055,8058,8122,8125],{"className":8048},[117],[17,8050],{"className":8051,"style":1031},[121],[17,8053,514],{"className":8054},[171],[17,8056],{"className":8057,"style":192},[191],[17,8059,8061],{"className":8060},[126,1035],[17,8062,8064,8114],{"className":8063},[142,143],[17,8065,8067,8111],{"className":8066},[147],[17,8068,8070,8090,8100],{"className":8069,"style":1045},[151],[17,8071,8072,8075],{"style":1048},[17,8073],{"className":8074,"style":1052},[159],[17,8076,8078],{"className":8077},[164,165,166,167],[17,8079,8081,8084,8087],{"className":8080},[171,167],[17,8082,69],{"className":8083,"style":173},[171,172,167],[17,8085,75],{"className":8086},[206,167],[17,8088,3026],{"className":8089},[171,167],[17,8091,8092,8095],{"style":1070},[17,8093],{"className":8094,"style":1052},[159],[17,8096,8097],{},[17,8098,564],{"className":8099},[126,1079,1080],[17,8101,8102,8105],{"style":1083},[17,8103],{"className":8104,"style":1052},[159],[17,8106,8108],{"className":8107},[164,165,166,167],[17,8109,1777],{"className":8110},[171,172,167],[17,8112,178],{"className":8113},[177],[17,8115,8117],{"className":8116},[147],[17,8118,8120],{"className":8119,"style":1105},[151],[17,8121],{},[17,8123],{"className":8124,"style":192},[191],[17,8126,8128,8205],{"className":8127},[171],[17,8129,8131,8134,8202],{"className":8130},[171],[17,8132],{"className":8133},[223,224],[17,8135,8137],{"className":8136},[77],[17,8138,8140,8194],{"className":8139},[142,143],[17,8141,8143,8191],{"className":8142},[147],[17,8144,8146,8172,8180],{"className":8145,"style":708},[151],[17,8147,8148,8151],{"style":240},[17,8149],{"className":8150,"style":244},[159],[17,8152,8154,8157,8160,8163,8166,8169],{"className":8153},[171],[17,8155,514],{"className":8156},[171],[17,8158,69],{"className":8159,"style":173},[171,172],[17,8161],{"className":8162,"style":635},[191],[17,8164,496],{"className":8165},[639],[17,8167],{"className":8168,"style":635},[191],[17,8170,326],{"className":8171},[171],[17,8173,8174,8177],{"style":259},[17,8175],{"className":8176,"style":244},[159],[17,8178],{"className":8179,"style":267},[266],[17,8181,8182,8185],{"style":270},[17,8183],{"className":8184,"style":244},[159],[17,8186,8188],{"className":8187},[171],[17,8189,326],{"className":8190},[171],[17,8192,178],{"className":8193},[177],[17,8195,8197],{"className":8196},[147],[17,8198,8200],{"className":8199,"style":3223},[151],[17,8201],{},[17,8203],{"className":8204},[304,224],[17,8206,8208,8211,8302],{"className":8207},[171],[17,8209],{"className":8210},[223,224],[17,8212,8214],{"className":8213},[77],[17,8215,8217,8294],{"className":8216},[142,143],[17,8218,8220,8291],{"className":8219},[147],[17,8221,8223,8272,8280],{"className":8222,"style":708},[151],[17,8224,8225,8228],{"style":240},[17,8226],{"className":8227,"style":244},[159],[17,8229,8231],{"className":8230},[171],[17,8232,8234,8237],{"className":8233},[171],[17,8235,72],{"className":8236},[171,172],[17,8238,8240],{"className":8239},[138],[17,8241,8243],{"className":8242},[142],[17,8244,8246],{"className":8245},[147],[17,8247,8249],{"className":8248,"style":5101},[151],[17,8250,8251,8254],{"style":5104},[17,8252],{"className":8253,"style":160},[159],[17,8255,8257],{"className":8256},[164,165,166,167],[17,8258,8260,8263,8266,8269],{"className":8259},[171,167],[17,8261,514],{"className":8262},[171,167],[17,8264,69],{"className":8265,"style":173},[171,172,167],[17,8267,496],{"className":8268},[639,167],[17,8270,326],{"className":8271},[171,167],[17,8273,8274,8277],{"style":259},[17,8275],{"className":8276,"style":244},[159],[17,8278],{"className":8279,"style":267},[266],[17,8281,8282,8285],{"style":270},[17,8283],{"className":8284,"style":244},[159],[17,8286,8288],{"className":8287},[171],[17,8289,326],{"className":8290},[171],[17,8292,178],{"className":8293},[177],[17,8295,8297],{"className":8296},[147],[17,8298,8300],{"className":8299,"style":298},[151],[17,8301],{},[17,8303],{"className":8304},[304,224],[10,8306,8307,8308,8478,8479,8643],{},"Let ",[17,8309,8311,8354],{"className":8310,"translate":21},[25],[17,8312,8314],{"className":8313},[29],[31,8315,8316],{"xmlns":33},[36,8317,8318,8351],{},[52,8319,8320,8323,8325],{},[60,8321,8322],{},"z",[64,8324,75],{},[77,8326,8327,8339],{},[52,8328,8329,8331,8333,8335,8337],{},[64,8330,319],{"stretchy":490},[60,8332,72],{},[64,8334,496],{},[492,8336,326],{},[64,8338,330],{"stretchy":490},[52,8340,8341,8343,8345,8347,8349],{},[64,8342,319],{"stretchy":490},[60,8344,72],{},[64,8346,507],{},[492,8348,326],{},[64,8350,330],{"stretchy":490},[105,8352,8353],{"encoding":107},"z = \\frac{(x+1)}{(x-1)}",[17,8355,8357,8376],{"className":8356,"ariaHidden":113},[112],[17,8358,8360,8363,8367,8370,8373],{"className":8359},[117],[17,8361],{"className":8362,"style":1789},[121],[17,8364,8322],{"className":8365,"style":8366},[171,172],"margin-right:0.04398em;",[17,8368],{"className":8369,"style":202},[191],[17,8371,75],{"className":8372},[206],[17,8374],{"className":8375,"style":202},[191],[17,8377,8379,8383],{"className":8378},[117],[17,8380],{"className":8381,"style":8382},[121],"height:1.53em;vertical-align:-0.52em;",[17,8384,8386,8389,8475],{"className":8385},[171],[17,8387],{"className":8388},[223,224],[17,8390,8392],{"className":8391},[77],[17,8393,8395,8466],{"className":8394},[142,143],[17,8396,8398,8463],{"className":8397},[147],[17,8399,8402,8428,8436],{"className":8400,"style":8401},[151],"height:1.01em;",[17,8403,8404,8407],{"style":4153},[17,8405],{"className":8406,"style":244},[159],[17,8408,8410],{"className":8409},[164,165,166,167],[17,8411,8413,8416,8419,8422,8425],{"className":8412},[171,167],[17,8414,319],{"className":8415},[223,167],[17,8417,72],{"className":8418},[171,172,167],[17,8420,507],{"className":8421},[639,167],[17,8423,326],{"className":8424},[171,167],[17,8426,330],{"className":8427},[304,167],[17,8429,8430,8433],{"style":259},[17,8431],{"className":8432,"style":244},[159],[17,8434],{"className":8435,"style":267},[266],[17,8437,8439,8442],{"style":8438},"top:-3.485em;",[17,8440],{"className":8441,"style":244},[159],[17,8443,8445],{"className":8444},[164,165,166,167],[17,8446,8448,8451,8454,8457,8460],{"className":8447},[171,167],[17,8449,319],{"className":8450},[223,167],[17,8452,72],{"className":8453},[171,172,167],[17,8455,496],{"className":8456},[639,167],[17,8458,326],{"className":8459},[171,167],[17,8461,330],{"className":8462},[304,167],[17,8464,178],{"className":8465},[177],[17,8467,8469],{"className":8468},[147],[17,8470,8473],{"className":8471,"style":8472},[151],"height:0.52em;",[17,8474],{},[17,8476],{"className":8477},[304,224],", then ",[17,8480,8482,8524],{"className":8481,"translate":21},[25],[17,8483,8485],{"className":8484},[29],[31,8486,8487],{"xmlns":33},[36,8488,8489,8521],{},[52,8490,8491,8493,8495],{},[60,8492,72],{},[64,8494,75],{},[77,8496,8497,8509],{},[52,8498,8499,8501,8503,8505,8507],{},[64,8500,319],{"stretchy":490},[60,8502,8322],{},[64,8504,507],{},[492,8506,326],{},[64,8508,330],{"stretchy":490},[52,8510,8511,8513,8515,8517,8519],{},[64,8512,319],{"stretchy":490},[60,8514,8322],{},[64,8516,496],{},[492,8518,326],{},[64,8520,330],{"stretchy":490},[105,8522,8523],{"encoding":107},"x = \\frac{(z-1)}{(z+1)}",[17,8525,8527,8545],{"className":8526,"ariaHidden":113},[112],[17,8528,8530,8533,8536,8539,8542],{"className":8529},[117],[17,8531],{"className":8532,"style":1789},[121],[17,8534,72],{"className":8535},[171,172],[17,8537],{"className":8538,"style":202},[191],[17,8540,75],{"className":8541},[206],[17,8543],{"className":8544,"style":202},[191],[17,8546,8548,8551],{"className":8547},[117],[17,8549],{"className":8550,"style":8382},[121],[17,8552,8554,8557,8640],{"className":8553},[171],[17,8555],{"className":8556},[223,224],[17,8558,8560],{"className":8559},[77],[17,8561,8563,8632],{"className":8562},[142,143],[17,8564,8566,8629],{"className":8565},[147],[17,8567,8569,8595,8603],{"className":8568,"style":8401},[151],[17,8570,8571,8574],{"style":4153},[17,8572],{"className":8573,"style":244},[159],[17,8575,8577],{"className":8576},[164,165,166,167],[17,8578,8580,8583,8586,8589,8592],{"className":8579},[171,167],[17,8581,319],{"className":8582},[223,167],[17,8584,8322],{"className":8585,"style":8366},[171,172,167],[17,8587,496],{"className":8588},[639,167],[17,8590,326],{"className":8591},[171,167],[17,8593,330],{"className":8594},[304,167],[17,8596,8597,8600],{"style":259},[17,8598],{"className":8599,"style":244},[159],[17,8601],{"className":8602,"style":267},[266],[17,8604,8605,8608],{"style":8438},[17,8606],{"className":8607,"style":244},[159],[17,8609,8611],{"className":8610},[164,165,166,167],[17,8612,8614,8617,8620,8623,8626],{"className":8613},[171,167],[17,8615,319],{"className":8616},[223,167],[17,8618,8322],{"className":8619,"style":8366},[171,172,167],[17,8621,507],{"className":8622},[639,167],[17,8624,326],{"className":8625},[171,167],[17,8627,330],{"className":8628},[304,167],[17,8630,178],{"className":8631},[177],[17,8633,8635],{"className":8634},[147],[17,8636,8638],{"className":8637,"style":8472},[151],[17,8639],{},[17,8641],{"className":8642},[304,224],". Substituting this into the above formula gives equation (4).",[10,8645,8646,8647,8675],{},"The advantage of this formula is that if we expand it to the ",[17,8648,8650,8663],{"className":8649,"translate":21},[25],[17,8651,8653],{"className":8652},[29],[31,8654,8655],{"xmlns":33},[36,8656,8657,8661],{},[52,8658,8659],{},[60,8660,1777],{},[105,8662,1777],{"encoding":107},[17,8664,8666],{"className":8665,"ariaHidden":113},[112],[17,8667,8669,8672],{"className":8668},[117],[17,8670],{"className":8671,"style":1789},[121],[17,8673,1777],{"className":8674},[171,172],"th term:",[17,8677,8679],{"className":8678,"translate":21},[20],[17,8680,8682,8829],{"className":8681,"translate":21},[25],[17,8683,8685],{"className":8684},[29],[31,8686,8687],{"xmlns":33,"display":34},[36,8688,8689,8826],{},[39,8690,8691],{"width":41},[43,8692,8693,8695,8819,8821],{},[46,8694],{"width":48},[46,8696,8697],{},[52,8698,8699,8701,8703,8705,8707,8709,8723,8737,8773,8775,8777],{},[60,8700,83],{},[64,8702,66],{},[60,8704,72],{},[64,8706,75],{},[492,8708,514],{},[560,8710,8711,8713,8721],{},[64,8712,564],{},[52,8714,8715,8717,8719],{},[60,8716,69],{},[64,8718,75],{},[492,8720,3026],{},[60,8722,1777],{},[77,8724,8725,8727],{},[492,8726,326],{},[52,8728,8729,8731,8733,8735],{},[492,8730,514],{},[60,8732,69],{},[64,8734,496],{},[492,8736,326],{},[516,8738,8739,8763],{},[52,8740,8741,8743,8761],{},[64,8742,319],{"fence":113},[77,8744,8745,8753],{},[52,8746,8747,8749,8751],{},[60,8748,72],{},[64,8750,507],{},[492,8752,326],{},[52,8754,8755,8757,8759],{},[60,8756,72],{},[64,8758,496],{},[492,8760,326],{},[64,8762,330],{"fence":113},[52,8764,8765,8767,8769,8771],{},[492,8766,514],{},[60,8768,69],{},[64,8770,496],{},[492,8772,326],{},[64,8774,496],{},[60,8776,1925],{},[52,8778,8779,8781,8817],{},[64,8780,319],{"fence":113},[516,8782,8783,8807],{},[52,8784,8785,8787,8805],{},[64,8786,319],{"fence":113},[77,8788,8789,8797],{},[52,8790,8791,8793,8795],{},[60,8792,72],{},[64,8794,507],{},[492,8796,326],{},[52,8798,8799,8801,8803],{},[60,8800,72],{},[64,8802,496],{},[492,8804,326],{},[64,8806,330],{"fence":113},[52,8808,8809,8811,8813,8815],{},[492,8810,514],{},[60,8812,69],{},[64,8814,496],{},[492,8816,326],{},[64,8818,330],{"fence":113},[46,8820],{"width":48},[46,8822,8823],{},[101,8824,8825],{},"(5)",[105,8827,8828],{"encoding":107},"\\ln x = 2\\sum_{k=0}^{n}\\frac{1}{2k+1}\\left(\\frac{x-1}{x+1}\\right)^{2k+1} + o\\left(\\left(\\frac{x-1}{x+1}\\right)^{2k+1}\\right) 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",[17,9357,9359,9417],{"className":9358,"translate":21},[25],[17,9360,9362],{"className":9361},[29],[31,9363,9364],{"xmlns":33},[36,9365,9366,9414],{},[52,9367,9368,9370],{},[60,9369,1925],{},[52,9371,9372,9374,9412],{},[64,9373,319],{"fence":113},[77,9375,9376,9388],{},[52,9377,9378,9380,9382,9384,9386],{},[64,9379,319],{"stretchy":490},[60,9381,72],{},[64,9383,507],{},[492,9385,326],{},[64,9387,330],{"stretchy":490},[52,9389,9390,9392,9394,9396,9398],{},[64,9391,319],{"stretchy":490},[60,9393,72],{},[64,9395,496],{},[492,9397,326],{},[516,9399,9400,9402],{},[64,9401,330],{"stretchy":490},[52,9403,9404,9406,9408,9410],{},[492,9405,514],{},[60,9407,69],{},[64,9409,496],{},[492,9411,326],{},[64,9413,330],{"fence":113},[105,9415,9416],{"encoding":107},"o\\left(\\frac{(x-1)}{(x+1)^{2k+1}}\\right)",[17,9418,9420],{"className":9419,"ariaHidden":113},[112],[17,9421,9423,9427,9430,9433],{"className":9422},[117],[17,9424],{"className":9425,"style":9426},[121],"height:1.8em;vertical-align:-0.65em;",[17,9428,1925],{"className":9429},[171,172],[17,9431],{"className":9432,"style":192},[191],[17,9434,9436,9443,9580],{"className":9435},[1015],[17,9437,9439],{"className":9438,"style":3136},[223,3135],[17,9440,319],{"className":9441},[3140,9442],"size2",[17,9444,9446,9449,9577],{"className":9445},[171],[17,9447],{"className":9448},[223,224],[17,9450,9452],{"className":9451},[77],[17,9453,9455,9568],{"className":9454},[142,143],[17,9456,9458,9565],{"className":9457},[147],[17,9459,9461,9531,9539],{"className":9460,"style":8401},[151],[17,9462,9464,9467],{"style":9463},"top:-2.6426em;",[17,9465],{"className":9466,"style":244},[159],[17,9468,9470],{"className":9469},[164,165,166,167],[17,9471,9473,9476,9479,9482,9485],{"className":9472},[171,167],[17,9474,319],{"className":9475},[223,167],[17,9477,72],{"className":9478},[171,172,167],[17,9480,496],{"className":9481},[639,167],[17,9483,326],{"className":9484},[171,167],[17,9486,9488,9491],{"className":9487},[304,167],[17,9489,330],{"className":9490},[304,167],[17,9492,9494],{"className":9493},[138],[17,9495,9497],{"className":9496},[142],[17,9498,9500],{"className":9499},[147],[17,9501,9504],{"className":9502,"style":9503},[151],"height:0.782em;",[17,9505,9507,9511],{"style":9506},"top:-2.786em;margin-right:0.0714em;",[17,9508],{"className":9509,"style":9510},[159],"height:2.5em;",[17,9512,9516],{"className":9513},[164,9514,9515,167],"reset-size3","size1",[17,9517,9519,9522,9525,9528],{"className":9518},[171,167],[17,9520,514],{"className":9521},[171,167],[17,9523,69],{"className":9524,"style":173},[171,172,167],[17,9526,496],{"className":9527},[639,167],[17,9529,326],{"className":9530},[171,167],[17,9532,9533,9536],{"style":259},[17,9534],{"className":9535,"style":244},[159],[17,9537],{"className":9538,"style":267},[266],[17,9540,9541,9544],{"style":8438},[17,9542],{"className":9543,"style":244},[159],[17,9545,9547],{"className":9546},[164,165,166,167],[17,9548,9550,9553,9556,9559,9562],{"className":9549},[171,167],[17,9551,319],{"className":9552},[223,167],[17,9554,72],{"className":9555},[171,172,167],[17,9557,507],{"className":9558},[639,167],[17,9560,326],{"className":9561},[171,167],[17,9563,330],{"className":9564},[304,167],[17,9566,178],{"className":9567},[177],[17,9569,9571],{"className":9570},[147],[17,9572,9575],{"className":9573,"style":9574},[151],"height:0.5324em;",[17,9576],{},[17,9578],{"className":9579},[304,224],[17,9581,9583],{"className":9582,"style":3136},[304,3135],[17,9584,330],{"className":9585},[3140,9442],", and as ",[17,9588,9590,9662],{"className":9589,"translate":21},[25],[17,9591,9593],{"className":9592},[29],[31,9594,9595],{"xmlns":33},[36,9596,9597,9659],{},[52,9598,9599,9619,9655,9657],{},[55,9600,9601,9608],{},[52,9602,9603,9606],{},[60,9604,9605],{},"lim",[64,9607,66],{},[52,9609,9610,9612,9615,9617],{},[60,9611,72],{},[64,9613,9614],{},"→",[64,9616,496],{},[60,9618,576],{"mathvariant":575},[516,9620,9621,9645],{},[52,9622,9623,9625,9643],{},[64,9624,319],{"fence":113},[77,9626,9627,9635],{},[52,9628,9629,9631,9633],{},[60,9630,72],{},[64,9632,507],{},[492,9634,326],{},[52,9636,9637,9639,9641],{},[60,9638,72],{},[64,9640,496],{},[492,9642,326],{},[64,9644,330],{"fence":113},[52,9646,9647,9649,9651,9653],{},[492,9648,514],{},[60,9650,69],{},[64,9652,496],{},[492,9654,326],{},[64,9656,75],{},[492,9658,326],{},[105,9660,9661],{"encoding":107},"\\lim_{x\\rightarrow+\\infty}\\left(\\frac{x-1}{x+1}\\right)^{2k+1} = 1",[17,9663,9665,9875],{"className":9664,"ariaHidden":113},[112],[17,9666,9668,9672,9727,9730,9866,9869,9872],{"className":9667},[117],[17,9669],{"className":9670,"style":9671},[121],"height:1.4923em;vertical-align:-0.4033em;",[17,9673,9675,9678],{"className":9674},[126],[17,9676,9605],{"className":9677},[126],[17,9679,9681],{"className":9680},[138],[17,9682,9684,9718],{"className":9683},[142,143],[17,9685,9687,9715],{"className":9686},[147],[17,9688,9691],{"className":9689,"style":9690},[151],"height:0.2583em;",[17,9692,9694,9697],{"style":9693},"top:-2.55em;margin-right:0.05em;",[17,9695],{"className":9696,"style":160},[159],[17,9698,9700],{"className":9699},[164,165,166,167],[17,9701,9703,9706,9709,9712],{"className":9702},[171,167],[17,9704,72],{"className":9705},[171,172,167],[17,9707,9614],{"className":9708},[206,167],[17,9710,496],{"className":9711},[171,167],[17,9713,576],{"className":9714},[171,167],[17,9716,178],{"className":9717},[177],[17,9719,9721],{"className":9720},[147],[17,9722,9725],{"className":9723,"style":9724},[151],"height:0.2083em;",[17,9726],{},[17,9728],{"className":9729,"style":192},[191],[17,9731,9733,9829],{"className":9732},[1015],[17,9734,9736,9742,9823],{"className":9735},[1015],[17,9737,9739],{"className":9738,"style":3136},[223,3135],[17,9740,319],{"className":9741},[3140,9515],[17,9743,9745,9748,9820],{"className":9744},[171],[17,9746],{"className":9747},[223,224],[17,9749,9751],{"className":9750},[77],[17,9752,9754,9811],{"className":9753},[142,143],[17,9755,9757,9808],{"className":9756},[147],[17,9758,9760,9780,9788],{"className":9759,"style":4150},[151],[17,9761,9762,9765],{"style":4153},[17,9763],{"className":9764,"style":244},[159],[17,9766,9768],{"className":9767},[164,165,166,167],[17,9769,9771,9774,9777],{"className":9770},[171,167],[17,9772,72],{"className":9773},[171,172,167],[17,9775,496],{"className":9776},[639,167],[17,9778,326],{"className":9779},[171,167],[17,9781,9782,9785],{"style":259},[17,9783],{"className":9784,"style":244},[159],[17,9786],{"className":9787,"style":267},[266],[17,9789,9790,9793],{"style":4176},[17,9791],{"className":9792,"style":244},[159],[17,9794,9796],{"className":9795},[164,165,166,167],[17,9797,9799,9802,9805],{"className":9798},[171,167],[17,9800,72],{"className":9801},[171,172,167],[17,9803,507],{"className":9804},[639,167],[17,9806,326],{"className":9807},[171,167],[17,9809,178],{"className":9810},[177],[17,9812,9814],{"className":9813},[147],[17,9815,9818],{"className":9816,"style":9817},[151],"height:0.4033em;",[17,9819],{},[17,9821],{"className":9822},[304,224],[17,9824,9826],{"className":9825,"style":3136},[304,3135],[17,9827,330],{"className":9828},[3140,9515],[17,9830,9832],{"className":9831},[138],[17,9833,9835],{"className":9834},[142],[17,9836,9838],{"className":9837},[147],[17,9839,9842],{"className":9840,"style":9841},[151],"height:1.089em;",[17,9843,9845,9848],{"style":9844},"top:-3.3029em;margin-right:0.05em;",[17,9846],{"className":9847,"style":160},[159],[17,9849,9851],{"className":9850},[164,165,166,167],[17,9852,9854,9857,9860,9863],{"className":9853},[171,167],[17,9855,514],{"className":9856},[171,167],[17,9858,69],{"className":9859,"style":173},[171,172,167],[17,9861,496],{"className":9862},[639,167],[17,9864,326],{"className":9865},[171,167],[17,9867],{"className":9868,"style":202},[191],[17,9870,75],{"className":9871},[206],[17,9873],{"className":9874,"style":202},[191],[17,9876,9878,9881],{"className":9877},[117],[17,9879],{"className":9880,"style":1358},[121],[17,9882,326],{"className":9883},[171],", the remainder term ensures that the error remains within the range of ",[17,9886,9888,9901],{"className":9887,"translate":21},[25],[17,9889,9891],{"className":9890},[29],[31,9892,9893],{"xmlns":33},[36,9894,9895,9899],{},[52,9896,9897],{},[492,9898,326],{},[105,9900,326],{"encoding":107},[17,9902,9904],{"className":9903,"ariaHidden":113},[112],[17,9905,9907,9910],{"className":9906},[117],[17,9908],{"className":9909,"style":1358},[121],[17,9911,326],{"className":9912},[171],[10,9914,9915,9916,9944,9945,9973,9974,10002],{},"However, this is still not sufficient, as the error of ",[17,9917,9919,9932],{"className":9918,"translate":21},[25],[17,9920,9922],{"className":9921},[29],[31,9923,9924],{"xmlns":33},[36,9925,9926,9930],{},[52,9927,9928],{},[492,9929,326],{},[105,9931,326],{"encoding":107},[17,9933,9935],{"className":9934,"ariaHidden":113},[112],[17,9936,9938,9941],{"className":9937},[117],[17,9939],{"className":9940,"style":1358},[121],[17,9942,326],{"className":9943},[171]," is still too large. According to the analysis of the remainder term above, the value of ",[17,9946,9948,9961],{"className":9947,"translate":21},[25],[17,9949,9951],{"className":9950},[29],[31,9952,9953],{"xmlns":33},[36,9954,9955,9959],{},[52,9956,9957],{},[60,9958,72],{},[105,9960,72],{"encoding":107},[17,9962,9964],{"className":9963,"ariaHidden":113},[112],[17,9965,9967,9970],{"className":9966},[117],[17,9968],{"className":9969,"style":1789},[121],[17,9971,72],{"className":9972},[171,172]," should be closer to ",[17,9975,9977,9990],{"className":9976,"translate":21},[25],[17,9978,9980],{"className":9979},[29],[31,9981,9982],{"xmlns":33},[36,9983,9984,9988],{},[52,9985,9986],{},[492,9987,326],{},[105,9989,326],{"encoding":107},[17,9991,9993],{"className":9992,"ariaHidden":113},[112],[17,9994,9996,9999],{"className":9995},[117],[17,9997],{"className":9998,"style":1358},[121],[17,10000,326],{"className":10001},[171]," for higher precision.",[10,10004,10005],{},"Considering the IEEE 754 floating-point standard, we know that any floating-point number is represented in the computer as:",[17,10007,10009],{"className":10008,"translate":21},[20],[17,10010,10012,10053],{"className":10011,"translate":21},[25],[17,10013,10015],{"className":10014},[29],[31,10016,10017],{"xmlns":33,"display":34},[36,10018,10019,10050],{},[52,10020,10021,10023,10025,10028,10030,10032,10034,10037],{},[60,10022,72],{},[64,10024,75],{},[492,10026,10027],{},"1.",[60,10029,72],{},[60,10031,72],{},[60,10033,72],{},[64,10035,10036],{},"×",[516,10038,10039,10041],{},[492,10040,514],{},[52,10042,10043,10046,10048],{},[60,10044,10045],{},"e",[60,10047,72],{},[60,10049,10],{},[105,10051,10052],{"encoding":107},"x = 1.xxx \\times 2^{exp}",[17,10054,10056,10074,10097],{"className":10055,"ariaHidden":113},[112],[17,10057,10059,10062,10065,10068,10071],{"className":10058},[117],[17,10060],{"className":10061,"style":1789},[121],[17,10063,72],{"className":10064},[171,172],[17,10066],{"className":10067,"style":202},[191],[17,10069,75],{"className":10070},[206],[17,10072],{"className":10073,"style":202},[191],[17,10075,10077,10081,10084,10088,10091,10094],{"className":10076},[117],[17,10078],{"className":10079,"style":10080},[121],"height:0.7278em;vertical-align:-0.0833em;",[17,10082,10027],{"className":10083},[171],[17,10085,10087],{"className":10086},[171,172],"xxx",[17,10089],{"className":10090,"style":635},[191],[17,10092,10036],{"className":10093},[639],[17,10095],{"className":10096,"style":635},[191],[17,10098,10100,10103],{"className":10099},[117],[17,10101],{"className":10102,"style":2590},[121],[17,10104,10106,10109],{"className":10105},[171],[17,10107,514],{"className":10108},[171],[17,10110,10112],{"className":10111},[138],[17,10113,10115],{"className":10114},[142],[17,10116,10118],{"className":10117},[147],[17,10119,10121],{"className":10120,"style":2590},[151],[17,10122,10123,10126],{"style":774},[17,10124],{"className":10125,"style":160},[159],[17,10127,10129],{"className":10128},[164,165,166,167],[17,10130,10132,10135,10138],{"className":10131},[171,167],[17,10133,10045],{"className":10134},[171,172,167],[17,10136,72],{"className":10137},[171,172,167],[17,10139,10],{"className":10140},[171,172,167],[10,10142,10143,10144,10178,10179,10217],{},"This means that we can directly obtain the exponent (",[17,10145,10147,10162],{"className":10146,"translate":21},[25],[17,10148,10150],{"className":10149},[29],[31,10151,10152],{"xmlns":33},[36,10153,10154,10159],{},[52,10155,10156],{},[101,10157,10158],{},"exp",[105,10160,10161],{"encoding":107},"\\text{exp}",[17,10163,10165],{"className":10164,"ariaHidden":113},[112],[17,10166,10168,10172],{"className":10167},[117],[17,10169],{"className":10170,"style":10171},[121],"height:0.625em;vertical-align:-0.1944em;",[17,10173,10175],{"className":10174},[171,315],[17,10176,10158],{"className":10177},[171],") and the fractional part (",[17,10180,10182,10202],{"className":10181,"translate":21},[25],[17,10183,10185],{"className":10184},[29],[31,10186,10187],{"xmlns":33},[36,10188,10189,10199],{},[52,10190,10191,10193,10195,10197],{},[492,10192,10027],{},[60,10194,72],{},[60,10196,72],{},[60,10198,72],{},[105,10200,10201],{"encoding":107},"1.xxx",[17,10203,10205],{"className":10204,"ariaHidden":113},[112],[17,10206,10208,10211,10214],{"className":10207},[117],[17,10209],{"className":10210,"style":1358},[121],[17,10212,10027],{"className":10213},[171],[17,10215,10087],{"className":10216},[171,172],") using bitwise operations.",[10,10219,10220,10221,10258,10259,10314,10315,10492],{},"Since the floating-point number is already in this form, we can directly use the fractional part ",[17,10222,10224,10243],{"className":10223,"translate":21},[25],[17,10225,10227],{"className":10226},[29],[31,10228,10229],{"xmlns":33},[36,10230,10231,10241],{},[52,10232,10233,10235,10237,10239],{},[492,10234,10027],{},[60,10236,72],{},[60,10238,72],{},[60,10240,72],{},[105,10242,10201],{"encoding":107},[17,10244,10246],{"className":10245,"ariaHidden":113},[112],[17,10247,10249,10252,10255],{"className":10248},[117],[17,10250],{"className":10251,"style":1358},[121],[17,10253,10027],{"className":10254},[171],[17,10256,10087],{"className":10257},[171,172],". If we calculate ",[17,10260,10262,10290],{"className":10261,"translate":21},[25],[17,10263,10265],{"className":10264},[29],[31,10266,10267],{"xmlns":33},[36,10268,10269,10287],{},[52,10270,10271,10273,10275,10277,10279,10281,10283,10285],{},[60,10272,83],{},[64,10274,66],{},[64,10276,319],{"stretchy":490},[492,10278,10027],{},[60,10280,72],{},[60,10282,72],{},[60,10284,72],{},[64,10286,330],{"stretchy":490},[105,10288,10289],{"encoding":107},"\\ln(1.xxx)",[17,10291,10293],{"className":10292,"ariaHidden":113},[112],[17,10294,10296,10299,10302,10305,10308,10311],{"className":10295},[117],[17,10297],{"className":10298,"style":622},[121],[17,10300,83],{"className":10301},[126],[17,10303,319],{"className":10304},[223],[17,10306,10027],{"className":10307},[171],[17,10309,10087],{"className":10310},[171,172],[17,10312,330],{"className":10313},[304]," using the earlier formula, it is equivalent to calculating ",[17,10316,10318,10356],{"className":10317,"translate":21},[25],[17,10319,10321],{"className":10320},[29],[31,10322,10323],{"xmlns":33},[36,10324,10325,10353],{},[52,10326,10327,10329,10331],{},[60,10328,83],{},[64,10330,66],{},[52,10332,10333,10335,10351],{},[64,10334,319],{"fence":113},[77,10336,10337,10339],{},[60,10338,72],{},[516,10340,10341,10343],{},[492,10342,514],{},[52,10344,10345,10347,10349],{},[60,10346,10045],{},[60,10348,72],{},[60,10350,10],{},[64,10352,330],{"fence":113},[105,10354,10355],{"encoding":107},"\\ln\\left(\\frac{x}{2^{exp}}\\right)",[17,10357,10359],{"className":10358,"ariaHidden":113},[112],[17,10360,10362,10366,10369,10372],{"className":10361},[117],[17,10363],{"className":10364,"style":10365},[121],"height:1.2em;vertical-align:-0.35em;",[17,10367,83],{"className":10368},[126],[17,10370],{"className":10371,"style":192},[191],[17,10373,10375,10381,10486],{"className":10374},[1015],[17,10376,10378],{"className":10377,"style":3136},[223,3135],[17,10379,319],{"className":10380},[3140,9515],[17,10382,10384,10387,10483],{"className":10383},[171],[17,10385],{"className":10386},[223,224],[17,10388,10390],{"className":10389},[77],[17,10391,10393,10475],{"className":10392},[142,143],[17,10394,10396,10472],{"className":10395},[147],[17,10397,10400,10450,10458],{"className":10398,"style":10399},[151],"height:0.6954em;",[17,10401,10402,10405],{"style":4153},[17,10403],{"className":10404,"style":244},[159],[17,10406,10408],{"className":10407},[164,165,166,167],[17,10409,10411],{"className":10410},[171,167],[17,10412,10414,10417],{"className":10413},[171,167],[17,10415,514],{"className":10416},[171,167],[17,10418,10420],{"className":10419},[138],[17,10421,10423],{"className":10422},[142],[17,10424,10426],{"className":10425},[147],[17,10427,10430],{"className":10428,"style":10429},[151],"height:0.5935em;",[17,10431,10432,10435],{"style":9506},[17,10433],{"className":10434,"style":9510},[159],[17,10436,10438],{"className":10437},[164,9514,9515,167],[17,10439,10441,10444,10447],{"className":10440},[171,167],[17,10442,10045],{"className":10443},[171,172,167],[17,10445,72],{"className":10446},[171,172,167],[17,10448,10],{"className":10449},[171,172,167],[17,10451,10452,10455],{"style":259},[17,10453],{"className":10454,"style":244},[159],[17,10456],{"className":10457,"style":267},[266],[17,10459,10460,10463],{"style":4176},[17,10461],{"className":10462,"style":244},[159],[17,10464,10466],{"className":10465},[164,165,166,167],[17,10467,10469],{"className":10468},[171,167],[17,10470,72],{"className":10471},[171,172,167],[17,10473,178],{"className":10474},[177],[17,10476,10478],{"className":10477},[147],[17,10479,10481],{"className":10480,"style":4198},[151],[17,10482],{},[17,10484],{"className":10485},[304,224],[17,10487,10489],{"className":10488,"style":3136},[304,3135],[17,10490,330],{"className":10491},[3140,9515],". According to the logarithmic formula:",[17,10494,10496],{"className":10495,"translate":21},[20],[17,10497,10499,10557],{"className":10498,"translate":21},[25],[17,10500,10502],{"className":10501},[29],[31,10503,10504],{"xmlns":33,"display":34},[36,10505,10506,10554],{},[52,10507,10508,10510,10512,10534,10536,10538,10540,10542,10544,10546,10548,10550,10552],{},[60,10509,83],{},[64,10511,66],{},[52,10513,10514,10516,10532],{},[64,10515,319],{"fence":113},[77,10517,10518,10520],{},[60,10519,72],{},[516,10521,10522,10524],{},[492,10523,514],{},[52,10525,10526,10528,10530],{},[60,10527,10045],{},[60,10529,72],{},[60,10531,10],{},[64,10533,330],{"fence":113},[64,10535,75],{},[60,10537,83],{},[64,10539,66],{},[60,10541,72],{},[64,10543,507],{},[101,10545,10158],{},[64,10547,10036],{},[60,10549,83],{},[64,10551,66],{},[492,10553,514],{},[105,10555,10556],{"encoding":107},"\\ln\\left(\\frac{x}{2^{exp}}\\right) = \\ln x - \\text{exp}\\times \\ln 2",[17,10558,10560,10695,10720,10742],{"className":10559,"ariaHidden":113},[112],[17,10561,10563,10567,10570,10573,10686,10689,10692],{"className":10562},[117],[17,10564],{"className":10565,"style":10566},[121],"height:1.836em;vertical-align:-0.686em;",[17,10568,83],{"className":10569},[126],[17,10571],{"className":10572,"style":192},[191],[17,10574,10576,10582,10680],{"className":10575},[1015],[17,10577,10579],{"className":10578,"style":3136},[223,3135],[17,10580,319],{"className":10581},[3140,9442],[17,10583,10585,10588,10677],{"className":10584},[171],[17,10586],{"className":10587},[223,224],[17,10589,10591],{"className":10590},[77],[17,10592,10594,10669],{"className":10593},[142,143],[17,10595,10597,10666],{"className":10596},[147],[17,10598,10600,10647,10655],{"className":10599,"style":5875},[151],[17,10601,10602,10605],{"style":240},[17,10603],{"className":10604,"style":244},[159],[17,10606,10608],{"className":10607},[171],[17,10609,10611,10614],{"className":10610},[171],[17,10612,514],{"className":10613},[171],[17,10615,10617],{"className":10616},[138],[17,10618,10620],{"className":10619},[142],[17,10621,10623],{"className":10622},[147],[17,10624,10627],{"className":10625,"style":10626},[151],"height:0.5904em;",[17,10628,10629,10632],{"style":5104},[17,10630],{"className":10631,"style":160},[159],[17,10633,10635],{"className":10634},[164,165,166,167],[17,10636,10638,10641,10644],{"className":10637},[171,167],[17,10639,10045],{"className":10640},[171,172,167],[17,10642,72],{"className":10643},[171,172,167],[17,10645,10],{"className":10646},[171,172,167],[17,10648,10649,10652],{"style":259},[17,10650],{"className":10651,"style":244},[159],[17,10653],{"className":10654,"style":267},[266],[17,10656,10657,10660],{"style":270},[17,10658],{"className":10659,"style":244},[159],[17,10661,10663],{"className":10662},[171],[17,10664,72],{"className":10665},[171,172],[17,10667,178],{"className":10668},[177],[17,10670,10672],{"className":10671},[147],[17,10673,10675],{"className":10674,"style":298},[151],[17,10676],{},[17,10678],{"className":10679},[304,224],[17,10681,10683],{"className":10682,"style":3136},[304,3135],[17,10684,330],{"className":10685},[3140,9442],[17,10687],{"className":10688,"style":202},[191],[17,10690,75],{"className":10691},[206],[17,10693],{"className":10694,"style":202},[191],[17,10696,10698,10702,10705,10708,10711,10714,10717],{"className":10697},[117],[17,10699],{"className":10700,"style":10701},[121],"height:0.7778em;vertical-align:-0.0833em;",[17,10703,83],{"className":10704},[126],[17,10706],{"className":10707,"style":192},[191],[17,10709,72],{"className":10710},[171,172],[17,10712],{"className":10713,"style":635},[191],[17,10715,507],{"className":10716},[639],[17,10718],{"className":10719,"style":635},[191],[17,10721,10723,10727,10733,10736,10739],{"className":10722},[117],[17,10724],{"className":10725,"style":10726},[121],"height:0.7778em;vertical-align:-0.1944em;",[17,10728,10730],{"className":10729},[171,315],[17,10731,10158],{"className":10732},[171],[17,10734],{"className":10735,"style":635},[191],[17,10737,10036],{"className":10738},[639],[17,10740],{"className":10741,"style":635},[191],[17,10743,10745,10748,10751,10754],{"className":10744},[117],[17,10746],{"className":10747,"style":364},[121],[17,10749,83],{"className":10750},[126],[17,10752],{"className":10753,"style":192},[191],[17,10755,514],{"className":10756},[171],[10,10758,10759],{},"Therefore:",[17,10761,10763],{"className":10762,"translate":21},[20],[17,10764,10766,10824],{"className":10765,"translate":21},[25],[17,10767,10769],{"className":10768},[29],[31,10770,10771],{"xmlns":33,"display":34},[36,10772,10773,10821],{},[52,10774,10775,10777,10779,10781,10783,10785,10787,10809,10811,10813,10815,10817,10819],{},[60,10776,83],{},[64,10778,66],{},[60,10780,72],{},[64,10782,75],{},[60,10784,83],{},[64,10786,66],{},[52,10788,10789,10791,10807],{},[64,10790,319],{"fence":113},[77,10792,10793,10795],{},[60,10794,72],{},[516,10796,10797,10799],{},[492,10798,514],{},[52,10800,10801,10803,10805],{},[60,10802,10045],{},[60,10804,72],{},[60,10806,10],{},[64,10808,330],{"fence":113},[64,10810,496],{},[101,10812,10158],{},[64,10814,10036],{},[60,10816,83],{},[64,10818,66],{},[492,10820,514],{},[105,10822,10823],{"encoding":107},"\\ln x = \\ln\\left(\\frac{x}{2^{exp}}\\right) + \\text{exp} \\times \\ln 2",[17,10825,10827,10851,10984,11005],{"className":10826,"ariaHidden":113},[112],[17,10828,10830,10833,10836,10839,10842,10845,10848],{"className":10829},[117],[17,10831],{"className":10832,"style":364},[121],[17,10834,83],{"className":10835},[126],[17,10837],{"className":10838,"style":192},[191],[17,10840,72],{"className":10841},[171,172],[17,10843],{"className":10844,"style":202},[191],[17,10846,75],{"className":10847},[206],[17,10849],{"className":10850,"style":202},[191],[17,10852,10854,10857,10860,10863,10975,10978,10981],{"className":10853},[117],[17,10855],{"className":10856,"style":10566},[121],[17,10858,83],{"className":10859},[126],[17,10861],{"className":10862,"style":192},[191],[17,10864,10866,10872,10969],{"className":10865},[1015],[17,10867,10869],{"className":10868,"style":3136},[223,3135],[17,10870,319],{"className":10871},[3140,9442],[17,10873,10875,10878,10966],{"className":10874},[171],[17,10876],{"className":10877},[223,224],[17,10879,10881],{"className":10880},[77],[17,10882,10884,10958],{"className":10883},[142,143],[17,10885,10887,10955],{"className":10886},[147],[17,10888,10890,10936,10944],{"className":10889,"style":5875},[151],[17,10891,10892,10895],{"style":240},[17,10893],{"className":10894,"style":244},[159],[17,10896,10898],{"className":10897},[171],[17,10899,10901,10904],{"className":10900},[171],[17,10902,514],{"className":10903},[171],[17,10905,10907],{"className":10906},[138],[17,10908,10910],{"className":10909},[142],[17,10911,10913],{"className":10912},[147],[17,10914,10916],{"className":10915,"style":10626},[151],[17,10917,10918,10921],{"style":5104},[17,10919],{"className":10920,"style":160},[159],[17,10922,10924],{"className":10923},[164,165,166,167],[17,10925,10927,10930,10933],{"className":10926},[171,167],[17,10928,10045],{"className":10929},[171,172,167],[17,10931,72],{"className":10932},[171,172,167],[17,10934,10],{"className":10935},[171,172,167],[17,10937,10938,10941],{"style":259},[17,10939],{"className":10940,"style":244},[159],[17,10942],{"className":10943,"style":267},[266],[17,10945,10946,10949],{"style":270},[17,10947],{"className":10948,"style":244},[159],[17,10950,10952],{"className":10951},[171],[17,10953,72],{"className":10954},[171,172],[17,10956,178],{"className":10957},[177],[17,10959,10961],{"className":10960},[147],[17,10962,10964],{"className":10963,"style":298},[151],[17,10965],{},[17,10967],{"className":10968},[304,224],[17,10970,10972],{"className":10971,"style":3136},[304,3135],[17,10973,330],{"className":10974},[3140,9442],[17,10976],{"className":10977,"style":635},[191],[17,10979,496],{"className":10980},[639],[17,10982],{"className":10983,"style":635},[191],[17,10985,10987,10990,10996,10999,11002],{"className":10986},[117],[17,10988],{"className":10989,"style":10726},[121],[17,10991,10993],{"className":10992},[171,315],[17,10994,10158],{"className":10995},[171],[17,10997],{"className":10998,"style":635},[191],[17,11000,10036],{"className":11001},[639],[17,11003],{"className":11004,"style":635},[191],[17,11006,11008,11011,11014,11017],{"className":11007},[117],[17,11009],{"className":11010,"style":364},[121],[17,11012,83],{"className":11013},[126],[17,11015],{"className":11016,"style":192},[191],[17,11018,514],{"className":11019},[171],[10,11021,11022,11023,11061],{},"This completes the calculation of ",[17,11024,11026,11043],{"className":11025,"translate":21},[25],[17,11027,11029],{"className":11028},[29],[31,11030,11031],{"xmlns":33},[36,11032,11033,11041],{},[52,11034,11035,11037,11039],{},[60,11036,83],{},[64,11038,66],{},[60,11040,72],{},[105,11042,395],{"encoding":107},[17,11044,11046],{"className":11045,"ariaHidden":113},[112],[17,11047,11049,11052,11055,11058],{"className":11048},[117],[17,11050],{"className":11051,"style":364},[121],[17,11053,83],{"className":11054},[126],[17,11056],{"className":11057,"style":192},[191],[17,11059,72],{"className":11060},[171,172],", equivalent to computing the logarithm of any base.",[10,11063,11064],{},"However, this is a general overview, and glibc employs various precision improvement techniques in its implementation.",{"title":11066,"searchDepth":11067,"depth":11067,"links":11068},"",2,[11069],{"id":418,"depth":11070,"text":11071},3,"Computation of ln⁡x\\ln xlnx",false,true,"md",null,{},"2023-12-07","\u002Fblog\u002F2023\u002F2023-12-07-how-do-computer-calculate-the-log-function",{"title":5,"description":12},"blog\u002F2023\u002F2023-12-07-How-do-computer-calculate-the-log-function","Analyzing the underlying implementation of the log function in computers from a numerical analysis perspective",[11083,11084],"algorithm","mathematics","Qro36Aew-9tjnllzdrEXSb_OJf_s1WOu0UZP_YFrcL0",1777090295699]