Equation with an exponential and a natural logarithm

Statement

Solve the equation, giving the result to two decimal places: $\ln\!\big(5+e^{2x}\big)=3.$

Solution

Exponentiating: $5+e^{2x}=e^{3}=20{.}0855$, hence $e^{2x}=15{.}0855$.

$2x=\ln 15{.}0855=2{.}71380\Rightarrow x=1{.}35690\approx 1{.}36$.

$\boxed{x\approx 1{.}36}$