Answer
See below.
Work Step by Step
We know that if $e^x=y$, then $ln y=x$, hence $2x=\ln(e^{2x}).$ We know that $\log_a x+\log_a y=\log_a (x\cdot y)$, hence $2x+\ln(1+e^{-2x})=\ln(e^{2x})+\ln(1+e^{-2x})=\ln(e^{2x}\cdot(1+e^{-2x}))=\ln(e^{2x}+e^{2x-2x})=\ln(e^{2x}+1).$
