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Solve this equation $4^{\log_2(x)}-2^{\log_2(x)}=3^{\log_3(12)}$

I thought to write $2^{\log_2(x)^2}-2^{\log_2(x)}=3^{\log_3(12)}$. Then is there a way to factorize $2^{\log_2(x)}$? I don't know how to proceed...

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  • $\begingroup$ Hint: $\log_n x$ is the inverse of $n^x$. $\endgroup$ – Henrik supports the community May 7 '15 at 15:28
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    $\begingroup$ Check basic logarithmic properties to untangle the exponents. In particular, $(a^2)^{\log_a x} = a^{2{\log_a x}} = a^{{\log_a {x^2}}} = x^2$ $\endgroup$ – Joffan May 7 '15 at 15:46
  • $\begingroup$ In general, $$\left(a^2\right)^b\ne a^{b^2}.$$ $\endgroup$ – Did May 7 '15 at 15:53
  • $\begingroup$ Hint: $n^{\log_n x}=x$ $\endgroup$ – steven gregory Feb 7 '17 at 21:04
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Let $\log_2 x = t$, then $2^{\log_2 (x)}=2^t=x$. Thus the given equation is $$x^2-2x=12$$

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  • $\begingroup$ why is it equal to 12? what about $3log_3(12)$? $\endgroup$ – anna May 7 '15 at 15:34
  • $\begingroup$ @anna Suppose $\log_3(12)=m$, then by definition $3^m=12$. Now consider $3^{\log_3(12)}=3^m=12$. $\endgroup$ – Anurag A May 7 '15 at 15:35
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Hint:

We have $a^{\log_a(b)}=b$ and we have the following:

$$4^{\log_2(x)}=2^{2\log_2(x)}=(2^{\log_2(x)})^2 = x^2$$

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