How do you simplify a log with an exponent in the base?
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.
Solution 1:
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 $$z=\frac{\log_x(y)}{b}$$
Solution 2:
You can calculate $$\frac{\ln(y)}{\ln(x^b)}=\frac{\ln(y)}{b\cdot \ln(x)}=\frac{\log_x(y)}{b}$$