# Question #64391

Sep 17, 2017

$t = \ln \left(\frac{13}{25}\right)$

#### Explanation:

We have: $25 \left(1 - {e}^{t}\right) = 12$

Divide both sides of the equation by $25$:

$R i g h t a r r o w 1 - {e}^{t} = \frac{12}{25}$

Subtract $1$ from both sides:

$R i g h t a r r o w - {e}^{t} = - \frac{13}{25}$

Multiply both sides by $- 1$:

$R i g h t a r r o w {e}^{t} = \frac{13}{25}$

Apply $\ln$ to both sides:

$R i g h t a r r o w \ln \left({e}^{t}\right) = \ln \left(\frac{13}{25}\right)$

Using the laws of logarithms:

$R i g h t a r r o w t \ln \left(e\right) = \ln \left(\frac{13}{25}\right)$

$R i g h t a r r o w t \cdot 1 = \ln \left(\frac{13}{25}\right)$

$\therefore t = \ln \left(\frac{13}{25}\right)$

Therefore, the solution to the equation is $\ln \left(\frac{13}{25}\right)$.