Solving $4^{32^x} = 16^{8^x}$ 
Solve for $x$:  $4^{32^x} = 16^{8^x}$.

So I have tried using log (so $x\log{4^{32}} = x\log{16^8}$), but that wasn't very helpful to me, and some random guessing gave me an answer of $1/2$, but I was wondering how I could be more mathematical…

Since$$
4^{32^x} = 2^{2 \cdot 32^x} = 2^{2 \cdot 2^{5x}} = 2^{2^{5x + 1}}
$$
and$$
16^{8^x} = 2^{4 \cdot 8^x} = 2^{4 \cdot 2^{3x}} = 2^{2^{3x + 2}},
$$
then$$
4^{32^x} = 16^{8^x} \Longleftrightarrow 2^{2^{5x + 1}} = 2^{2^{3x + 2}} \Longleftrightarrow 2^{5x + 1} = 2^{3x + 2}\\\Longleftrightarrow 5x + 1 = 3x + 2 \Longleftrightarrow x = \frac{1}{2}.
$$
