It is often useful to be able to take logs of something raised to a power or to move a number multiplying a log inside the log. To do this the rule below can be used:

log_ab^c = c log_ab

For example,

log(9) = log(3²) = 2 log(3)

This can be used to either take a power outside of a log so it becomes a multiplication that is easier to work with or take a multiplication inside a log so you are only working with logs.

Proof when c is an integer

When c in the above equation is a positive integer it is simple. Since b^c means b multiplied by itself c times we can use the rules of log addition to get
log(b^c) = log(b….b) = log(b) + log(b) + … + log(b) = clog(b)

Similarly when c is negative we divide 1 by b c times so with using the rules of log subtraction we get
log(b^-c = log(1/b …. 1/b) = -log(b) – … -log(b) = -clog(b)

Proof when c is a real number

Noticing that b = a^log_ab
We can write
b^c = (a^log_ab)^c=a^clog_ab
So taking log a of both sides gives us
log(b^c) = c log_ab as required

Notice that this uses the fact that a log and a power cancel.
Ie) log_aa^b = b
and a^log_ab = b