Your starting equation is
2^{16^x} = 128*4^x.
Notice that 2^(16^x) = 2^(2^(4x)).
So, the original equation is equivalent to
2^(2^(4x)) = 2^7 * 2^(2x).
It implies for indexes
2^(4x) = 2x + 7.
This equation can not be solved algebraically, but it can be solved approximately
with reasonable precision using numerical methods and special solvers in the Internet.
I used online solver at the site www.desmos.com/calculator. The calculator produced two approximate solutions
= -3.49997 (approximately), and = 0.77389 (approximately)
As the reference to the solver' solution, see this link
https://www.desmos.com/calculator/tmxp2wpqt3
Therefore, may have two values
= 0.08839 (approx.) and = 1.709874 (approx.)
These values, = 0.08838 and = 1.709874, are your ANSWER to the problem's question.
