# How do you solve for x?: e^(-2x) = 1/3?

Oct 23, 2015

#### Answer:

$x = \ln \frac{3}{2} \approx 0.54930614433$

#### Explanation:

Convert the equation to logarithm form.

$\textcolor{w h i t e}{X X} {e}^{- 2 x} = \frac{1}{3}$

$\textcolor{w h i t e}{X X} \Leftrightarrow {\log}_{e} \left(\frac{1}{3}\right) = - 2 x$

$\textcolor{w h i t e}{X X} \ln \left(\frac{1}{3}\right) = - 2 x$

Isolate x.

$\textcolor{w h i t e}{X X} \left(- \frac{1}{2}\right) \left[\ln \left(\frac{1}{3}\right)\right] = \left(- \frac{1}{2}\right) \left(- 2 x\right)$

$\textcolor{w h i t e}{X X} \frac{- \ln \left(\frac{1}{3}\right)}{2} = x$

Simplify.

$\textcolor{w h i t e}{X X} \ln \frac{{\left(\frac{1}{3}\right)}^{-} 1}{2} = x$ Theorem: Logarithm of a Power

$\textcolor{w h i t e}{X X} \textcolor{red}{x = \ln \frac{3}{2}}$

You can leave the answer at that since you can't really get $\ln \left(3\right)$ without a calculator. The actual answer is around 0.54930614433.