What is the answer?

Let's stick to log properties...use log[m]n to represent log-base-m-of-n, and ln is the natural log, i.e., base e.

log[4]32 = ln32/ln4
log[4]32 = ln(2^5)/ln(2^2)
log[4]32 = 5ln2/2ln2
log[4]32 = (5/2)(ln2/ln2)
log[4]32 = 5/2

log 32, base 4 = x

Changing from log form to equivalent exponent form we get:

4^x = 32

Then changing to prime bases on each side we get

(2^2)^x = 2^5 which is the same as 2^(2x) = 2^5

Now since the bases are (cooperatively) the same on each
side, the exponents must be the same: i.e. 2x = 5

or x = 5/2

[Were the prime bases NOT the same on each side then you would "force" the same base, e or 10 on each side -- and you'd probably require a calculator.]

Write as x = log32 to base 4
so 4^x = 32
so 2^2x = 2^5
so 2x = 5
so x = 2.5
so log 32 = 2.5 to base 4.

x = 2.5
4^5/2 = 32.

4^x=32
2^2x=2^5
2x=5
x=5/2

log32 base 4=5/2