Answer
The value of t is 35.065.
Work Step by Step
Consider the exponential equation.
${{e}^{-0.1t}}=0.03$
Simplify the exponent ${{e}^{-0.1t}}=0.03$ as follows.
${{e}^{-0.1t}}=0.03$
Take the natural logarithm on both sides,
$\begin{align}
& -0.1t=\ln 0.03 \\
& t=\frac{\ln 0.03}{-0.1}
\end{align}$
Use a calculator and solve as:
$\begin{align}
& t\approx \frac{-3.5065}{-0.1} \\
& =35.065
\end{align}$
Check:
Substitute $t=35.065$ in the given equation.
$\begin{matrix}
{{e}^{-0.1\left( 35.065 \right)}}\overset{?}{\mathop{=}}\,0.03 \\
\text{ }{{e}^{-3.5065}}\overset{?}{\mathop{=}}\,0.03 \\
\text{ }0.03=0.03 \\
\end{matrix}$
Thus, the obtained solution is correct.
