Logarithmic Equation with Two Variables

We have $\log_4(mn) = \log_2(m+n) \implies \frac{1}{2}\log_2(mn) = \log_2(m+n) \implies \sqrt{mn} = m+n$. Squaring both sides gives $mn = m^2 + 2mn + n^2 \implies m^2 + mn + n^2 = 0$. Since $m$ and $n$ are positive real numbers, $m^2 + mn + n^2 > 0$ always. Thus, no positive real values of $m$ and $n$ can satisfy the equation.