SIMPLIFY THE FOLLOWING EXPRESSION: Z=LOG(E^5Y)

i MOVED THE EXPONENT TO THE FRONT:
Z=5YLOGE IS THIS CORRECT?

yes, as far as you went.

But, if you are using natural logs, log(e) = 1, so the answer is just 5y

By definition

logbb^n = n
b^(logbn) = n

because logs and powers are inverse functions, just like √x and x^2 are inverses.

Like + and - or * and /

It's just that we don't have any special symbols for the notation.