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$4^x=2^x+6$

Given that $x$ is in the form "log base $a$ of $b$" and both $a$ and $b$ are prime numbers, what is the ordered pair $(a,b)$:

I have no idea how to solve this, I've been staring at it for hours now and no matter what I do I cannot seem to find a way to arrive at the solution. After simply plugging in numbers I realized that the answer is $(2,3)$ but I would like to know the actual way to solve it. Any help would be greatly appreciated!

N. F. Taussig
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bbuddy27
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1 Answers1

10

Hint: Since $4^x=(2^x)^2$, you can write your equation as $$(2^x)^2-2^x-6=0.$$ Now if you set $u=2^x$ you get a quadratic!

Workaholic
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