e(x) = 1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! + (x^5)/5! + ...

for logs:

Euler's method:

log(sqrt(x*y)) = (log(x)+log(y))/2  (all logs to the same base)
essentially, binary search
eg, for log_10, consider log(64):
10<64<100, so log(10)=1 < log(64) < log(100)=2
log(sqrt(1000)=31.6227766+)=1.5
31.etc < 64 < 100 so log(31.etc)=1.5 < log(64) < log(100)=2

Taylor series

ln(x) = sum(n=1..infty) (-1)^(n+1) [(x-1)^n / n]  (convergent for 0<x<=2)
To bring numbers into range, consider
log_a(b) = ln(b)/ln(a) = -ln(1/b)/-ln(1/a) = ln(1/b)/ln(1/a)
but, the farther from 1 x is, the slower the above converges.
s = f / (2+f), because then log(1+f) = log(1+s) - log(1-s)
if f is small, s will be about half f
if f is large, s will be about 1