Question

Solve the equation.

$e^{-2x}=\frac{1}{3}$

Short answer

The solution to the equation, $e^{-2x}=\frac{1}{3}$is, $x=-\frac{1}{2}\ln \frac{1}{3}$.

Step-by-step solution

Step 1. Given Information

The given equation to be solved is $e^{-2x}=\frac{1}{3}$.

Step 2. Formula used

Natural logarithm:

$\ln e=1$

Step 3. Finding Solution

The given equation is $e^{-2x}=\frac{1}{3}$.

Take logarithm on both sides.

$$\begin{gathered} \ln e^{-2x}=\ln \frac{1}{3} \\ -2x\ln e=\ln \frac{1}{3} \\ -2x=\ln \frac{1}{3} \\ x=-\frac{1}{2}\ln \frac{1}{3} \end{gathered}$$

The solution to the given equationis,$x=-\frac{1}{2}\ln \frac{1}{3}$.