Logarithm Powers Proof?

Question

Prove that log a^b = b log a

Answer 1

The left hand side of the eqn is

log(a^b) = log(a*a*...*a) where there are "b" a's multiplied by each other in the argument.

Using the property of logarithms that log(xy) = log(x) + log(y) to the above expression gives

log(a*a*...*a)
= log(a) + log(a) + ... + log(a) where there are "b" log(a)'s being summed. We can simplify this of course as (b)log(a) and we are done.

Answer 2

a = 10^log(a)

a ^ b = [10^log(a)] ^ b ; substitute from previous line

a ^ b = 10^(b log (a)) ; because (a^m)^n = a^(mn)

take log of both sides

log (a^b) = b log(a)