Answer
$$b = \ln (e^a+2)-\ln (3)$$
Work Step by Step
From the given figure , we have
\begin{aligned}
\text{Area of } \ A&= \int_0^ae^xdx\\
&= e^x\bigg|0^a\\
&= e^a-1\\
\text{Area of } \ B&= \int_0^be^xdx\\
&= e^x\bigg|0^b\\
&= e^b-1\\
\end{aligned}
Since area of B is three times the area of A
\begin{aligned}
\text{Area of } \ B &=3\text{Area of } \ A\\
3e^b-3&=e^a-1\\
3e^b &=e^a+2\\
e^b &=\frac{1}{3}(e^a+2)\\
b &= \ln \frac{e^a+2}{3}\\
&= \ln (e^a+2)-\ln (3)
\end{aligned}
