Solve e^-0.08t=0.27

$$e ^ { - 0.08 t } = 0.27$$

Solve for t

$$t=-\frac{75\ln(3)}{2}+25\ln(10)\approx 16.3666665$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$e^{-0.08t}=0.27$$

Take the logarithm of both sides of the equation.

$$\log(e^{-0.08t})=\log(0.27)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$-0.08t\log(e)=\log(0.27)$$

Divide both sides by \log(e).

$$-0.08t=\frac{\log(0.27)}{\log(e)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$-0.08t=\log_{e}\left(0.27\right)$$

Divide both sides of the equation by -0.08, which is the same as multiplying both sides by the reciprocal of the fraction.

$$t=\frac{\ln(\frac{27}{100})}{-0.08}$$