What are the parts of a logarithm called?

I would just call it the argument, it makes sense of thinking of $\log_x$ as an operator, which is applied to an argument. So I would say that $y$ is the argument for the operator $\log_x$ when looking at the expression $\log_x(y)$.

In the days when people used logithm tables, the integer part was the characteristic, and the decimal was the mantissa.

So $\log 20 = 1.30103$, makes 1 the characteristic (the bit after E...)and 0.30103 the mantissa (which the log tables tell you).

In $b^n = x$ or $\operatorname{lg}_b x = n$, b is the base, and n is the exponent, x is the argument of the function.

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.