# Can someone help me solve for x? (Exponential Equation)

## $\frac{50}{1 + {e}^{-} x} = 4$

Apr 13, 2017

$x = - \ln 46$

#### Explanation:

Given:

$\frac{50}{1 + {e}^{- x}} = 4$

Multiply both sides by $1 + {e}^{- x}$ to get:

$50 = 4 + 4 {e}^{- x}$

Subtract $4$ from both sides to get:

$46 = {e}^{- x}$

Take natural logs of both sides to get:

$\ln 46 = - x$

Multiply both sides by $- 1$ to get:

$- \ln 46 = x$

That is:

$x = - \ln 46$

(which is the same as $\ln \left(\frac{1}{46}\right)$)

Apr 13, 2017

$x = - 2.44 \text{ to 2 dec. places}$

#### Explanation:

$\textcolor{b l u e}{\text{cross-multiply}}$ the equation.

$\Rightarrow 4 \left(1 + {e}^{-} x\right) = 50$

$\text{divide both sides by 4}$

$\frac{\cancel{4} \left(1 + {e}^{-} x\right)}{\cancel{4}} = \frac{50}{4}$

$\Rightarrow 1 + {e}^{-} x = 12.5$

$\text{subtract 1 from both sides}$

$\cancel{1} \cancel{- 1} + {e}^{-} x = 12.5 - 1$

$\Rightarrow {e}^{-} x = 11.5$

$\text{using "color(blue)"law of logarithms}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{\log {x}^{n} \Leftrightarrow n \log x} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{take ln (natural log ) of both sides}$

$\ln {e}^{-} x = \ln 11.5$

$\Rightarrow - x {\cancel{\ln e}}^{1} = \ln 11.5$

$\Rightarrow x = - \ln 11.5$

$\Rightarrow x \approx - 2.44 \text{ to 2 dec. places}$