Page 33 - Standard Handbook Of Petroleum & Natural Gas Engineering
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22 Mathematics
log(a/b) = log a - log b
log(l/n) = -log n
log(a") = n log a
log,(b) = 1, where b is either 10 or e
log 0 = -03
log 1 = 0
log,,e = M = 0.4342944819 . . . , so for conversion
loglox = 0.4343 logex
and since 1/M = 2.302585, for conversion (In = loge)
In x = 2.3026 loglox
Binomial Theorem
Let
n, = n
n(n - 1)
n2 = -
2!
n(n - l)(n - 2)
n3 =
3!
etc. Then for any n, 1x1 < 1,
(1 + x)" = 1 + n,x + n2x2 + ngx3 + . . .
If n is a positive integer, the system is valid without restriction on x and
completes with the term n,x".
Some of the more useful special cases are the following [l]:
1 1 1 5
Jl+X=(l+x)'/Z =1+-x--x~+-xxJ--x4+ ... (Ixl< 1)
2 8 16 128
1 1 5 10
G'Y = (1 + x)q3 = 1 +-x--x2 --x3 --x4 + ... (Ixl< 1)
3 9 81 243
-- -(1+x)-'=1-x+x2-xxJ+x4- . . . (Ixl< 1)
l+x
