However, your exponent laws need work!
(1/2)^(5t) / (1/2)^5 = (1/2) ^ (5t-5) is not (1/2)^t.

Instead of subtracting 5 from (5t), you need to divide, in the exponent. That means you want roots:
Fifth-root[(1/2)^(5t)] = [(1/2)^(5t)]^(1/5) = (1/2)^t

Once you have that (1/2)^t = [3.1^(-3)]^(1/5)
you other idea is fine:

Logarithm base (1/2) of {[3.1^(-3)]^(1/5)}

[Log Definition: If A^B=C, then B = Log base A of C.]

Just a small bit of weirdness, to calculate Log base A of C on most calculators, you must enter:
Log(C)/Log(A)

So your answer is: t = Log((3.1^-3)^0.2)/Log(0.5)
which is correct: calculate (1/2)^(5*t) and compare with 3.1^-3
Notice that none of the given answers on Prepzel is correct!

It appears that the question was mistyped:
Instead of 3.10^-3, it should have been 3×10^-3
(scientific notation for 0.003)