What are the parts of a logarithm called? $$\log_x y = z$$
$x$ is the base.
$z$ is the exponent or power.
What's $y$ called?

As I learned it, $y$ in your equation is the "power."
$z$ is very sharply the "exponent" (or "logarithm"), not the power.  However, I also learned that few people make this sharp of a distinction.
"The fifth power of two" is equal to $32$.  Is thirty-two an exponent?  Of course not.
Is it a power?  Well, I just said so in the question, didn't I?
Which power is it?  The fifth power of two, of course.  $5$ isn't the power.  It's which power (of what base) is being referred to.
$2$ is the base.  $5$ is the exponent.  $32$ is the power.
You can retain these words regardless of whether the equation you reference is a logarithm or exponentiation.
