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

Question

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...

Answer 1

Let $\log_2 x = t$, then $2^{\log_2 (x)}=2^t=x$. Thus the given equation is $$x^2-2x=12$$

Answer 2

Hint:

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

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