Solving $2^{16^x} = 16^{2^x}$ 
Find $x$ if $2^{16^x} = 16^{2^x}$.

If $x = 0$ we have $2=16$.
$8$ times or $(2^3)$ difference and if $x = 1$ we have $65536=256$.
$256$ or $2^8$ difference. I can't see anything useful in figuring this out.
I have a feeling $x$ must be a fraction. I know only know about logarithm. I haven't learned anything about the number $e$ or $ln$ yet.

$16^{2^x}=(2^4)^{2^x}=2^{4\cdot 2^x}=2^{2^2\cdot 2^x}=2^{2^{x+2}}$. So
$$2^{16^x}=16^{2^x}\iff 2^{16^x}=2^{2^{x+2}}\iff$$ $$\iff 16^x=2^{x+2}\iff 2^{4x}=2^{x+2} \iff 4x=x+2 \iff x=\frac{2}{3}.
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