Proof of $e^{\ln(x)\ln(2)}$, which natural logarithm do I bring down?

Question

I'm currently stumped with the proof for the following problem:

$$F(x) = 2^{\ln(x)}$$ $$\Rightarrow F(x) = y$$ $$y = 2^{\ln(x)}$$ $$\ln(y) = \ln(2^{\ln(x)})$$ $$\ln(y) = \ln(x)\cdot\ln(2)$$ $$y = e^{\ln(x)\cdot\ln(2)}$$

Now, how do I know which natural logarithm to bring down? What ensures that my answer must be $2^{\ln(x)}$ instead of $x^{\ln(2)}$?

Thanks! :)

Answer 1

Both are the same: $$ 2^{\log(x)}=e^{\log(x)\log(2)}=x^{\log(2)} $$