TITLE: How do you simplify a log with an exponent in the base?
QUESTION [12 upvotes]: Suppose you had:
$$\log_{x^b}(y)$$
How can you simplify this? Do you use the change of base formula?
Note: I tried to come up with something similar to a homework problem without actually being a homework problem. I think this is the most simple form.

REPLY [25 votes]: We have your original function
$$\log_{x^b}(y)=z$$
Following basic rules for logarithms, assuming $x,y,z>0$
$$(x^b)^z=y$$
$$x^{bz}=y$$
$$\log_x(y)=bz$$
Thus $z$ can be expressed as
$$z=\frac{\log_x(y)}{b}$$

REPLY [22 votes]: You can calculate $$\frac{\ln(y)}{\ln(x^b)}=\frac{\ln(y)}{b\cdot \ln(x)}=\frac{\log_x(y)}{b}$$