Question 1209868
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If 2^{16^x} = 128*4^x, then find 2^x.
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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

    {{{x[1]}}} = -3.49997  (approximately),  and   {{{x[2]}}} = 0.77389  (approximately)


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;As the reference to the solver' solution, see this link

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;https://www.desmos.com/calculator/tmxp2wpqt3



Therefore, {{{2^x}}}  may have two values

    {{{2^(-3.49997)}}} = 0.08839  (approx.)  and   {{{2^0.77389}}} = 1.709874 (approx.)


These values, {{{x[1]}}} = 0.08838  and  {{{x[2]}}} = 1.709874,  are your <U>ANSWER</U>  to the problem's question.
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Solved.