[Math] How to Solve for $x$ in a Particular Exponential Equation

algebra-precalculusexponential functionexponentiation

I am trying to solve for $x$ in

$x^2=(16)^{2x}.$

So I started this way:
I took square root of both sides and got

$x=16^x$

Then I took the logarithm of both sides and got
$\log x=x \log 16.$

This is where I got stuck.

Best Answer

Contrary to some other answers,

$$x=-0.36424988978364795656$$ is the real solution. (Can be expressed in terms of the Lambert $W$ function; otherwise there is no analytical expression.)

https://www.wolframalpha.com/input/?i=x%5E2%3D256%5Ex

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