Answer
$x=2-log{\frac{5}{2}}$
Work Step by Step
We know that by definition if $y=a^x$, then $log_a {y}=x$ , also $log_e x=ln x$, hence if $y=e^x$, then $ln{e^x}=log_e {e^x}=x$ and vice versa and that $ln{\frac{1}{x}}=-ln{x}$.
Hence if $2\cdot10^{2-x}=5$, then $10^{2-x}=\frac{5}{2}$. Solve the equation above to obtain (after taking $log$ of both sides): \begin{align*}2-x=log{\frac{5}{2}}\end{align*} \begin{align*}-x=log{\frac{5}{2}}-2\end{align*} \begin{align*}x=2-log{\frac{5}{2}}\end{align*}
