A problem involving the distance between points on two logarithmic curves

Question

Let the curves $\Gamma_1$ and $\Gamma_2$ respectively represent $y = \log_2x$ and $\log_4x$.

Let the line $y=k$ intersect $\Gamma_1$ and $\Gamma_2$ respectively at the points $P$ and $Q$. If $\overline{PQ} = 20$, what are the $x$ coordinates of $P$ and $Q$?

I know that $\log_4x = \frac12\log_2x$ and that the curve grows slower than $\log_2x$, but I think I'm missing something here because I don't see any solution. :P

Answer 1

We get $$k=\log_2{x}$$ if $$x_1=e^{k\ln(2)}$$ and $$x_2=e^{k\ln(4)}$$

Answer 2

$\log_4 k = \log_2 (k-20)$

$\rightarrow \ln 4 \ln (k-20) = \ln 2 \ln k $

$\rightarrow 2 \ln (k-20) = \ln k $

$\rightarrow (k-20)^2 = k$

$\rightarrow k = 25$