Answer
Proof.
Work Step by Step
The exponential growth function as, $P\left( T \right)={{P}_{0}}{{e}^{kT}},\ \ \ k>0$
Now, suppose at any time $T$, $P\left( T \right)$ is double of ${{P}_{0}}$, so put $P\left( T \right)=2{{P}_{0}}$ in the exponential growth function:
$\begin{align}
& P\left( T \right)={{P}_{0}}{{e}^{kT}} \\
& 2{{P}_{0}}={{P}_{0}}{{e}^{kT}} \\
& 2={{e}^{kT}}
\end{align}$
Taking ln on both sides:
$\begin{align}
& 2={{e}^{kT}} \\
& \ln 2=\ln {{e}^{kT}} \\
& \ln 2=kT \\
& \frac{\ln 2}{k}=T
\end{align}$
Therefore, the doubling time $T$ is given by $T=\frac{\ln 2}{k}$ for the exponential growth at rate $k$.
