Answer
$$\overline f = \frac{3}{{\ln 4}}$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = {e^{2x}}{\text{ on the interval }}\left[ {0,\ln 2} \right] \cr
& {\text{Find the average value using }}\overline f = \frac{1}{{b - a}}\int_a^b {f\left( x \right)} dx \cr
& \overline f = \frac{1}{{\ln 2 - 0}}\int_0^{\ln 2} {{e^{2x}}} dx \cr
& {\text{Integrate}} \cr
& \overline f = \frac{1}{{\ln 2}}\left[ {\frac{1}{2}{e^{2x}}} \right]_0^{\ln 2} \cr
& \overline f = \frac{1}{{\ln 4}}\left[ {{e^{2\ln 2}} - {e^0}} \right] \cr
& \overline f = \frac{1}{{\ln 4}}\left[ {4 - 1} \right] \cr
& \overline f = \frac{3}{{\ln 4}} \cr
& \cr
& {\text{Graph}} \cr} $$
