500
Hence prove that(7)Exponential and logarithmic functionslog11x=x+2+3+3+...when -1 S x < 1 and that(8) log2=1+ +3-1 ...
Show that replacing x by (-x) in (7) gives the formula(^34) ...
(9) log(1+x) =x- 2 +3
_X+
and that addition gives the formula
(10) log1 +x = 2 x +3 + S + 7 { .. .l
which holds when (xi < I. Use (10) and a little imagination to obtain the
formula
(11) log2=2(3+33+535+. .l
from which we could, with the aid of a calculator or computer, calculate log 2
correct to many decimal places. Some persons need approximations as good as
that in
(12) log 2 = 0.69314 71805 59945 30941.
If we want log 10, we could get log 8 by multiplying (12) by 3 and then add the
result to log (V) which can be calculated from the formula
(13) log
8
= 2(4+393+595+7.9'+
in which the series converges quite rapidly. The result is
(14) log 10 = 2.30258 50929 94045 68402.
To get log 3, we could let x = l in (9), but this series converges rather slowly. It
is much better to calculate log 9 and then log 3 from (14) and the formula
(15) log
10
9 = 2 (
1
19 +
1
3.193 +5 195 +
1
This gives
(16) log 3 = 1.09861 22886 68109 69140.
14 Elementary combinations of the logarithms of 2, 3, and 10 give the loga-
rithms of 2, 3, 4, 5, 6, 8, 9, 10, but 7 is missing. Show how the proximity of
49 to SO enables us to calculate log 7 with the aid of a series that converges
rapidly.
15 This book does not recommend learning a formula for dy/dx when y =
f(x)D(2), where f and g are given functions. If (as frequently happens in good
old-fashioned mathematics examinations) we are required to produce the deriva-