Towering $x$

$\begin{aligned} 2^{16^x} &= 16^{2^x}\\ 2^{2^{4x}} &= 2^{4 \times 2^x}\\ 2^{4x} &= 2^{x+2}\\ 4x&=x+2\\ x&=\dfrac{2}{3} \end{aligned}$

Let us simplify the RHS as:

( 16^{2^x} = {2^4}^{2^x} = {2^{2^2}}^{2^x} = 2^{2^{x+2}} )\ (i).

Equating (i) with the LHS now gives:

2^(16^x) = 2^(2^(x+2));

or 16^x = 2^(x+2);

or (2^4)^x = 2^(x+2);

or 2^(4x) = 2^(x+2);

or 4x = x + 2;

or 3x = 2;

or x = 2/3.

$\large 2^{16^x} =16^{2^x}$

$\large 2^{16^x} = 2^{4 \times 2^x}$

$\large 16^x = 4 \times 2^x$

$\large (2^4)^x = 2^{2+x}$

$\large 4x = 2+x$

$\large \therefore x = \dfrac 23$