Solve 75e^0.1t=10

$$75 e ^ { 0.1 t } = 10$$

Solve for t

$$t=\ln(\frac{1024}{576650390625})\approx -20.149030205$$

Steps Using Definition of Logarithm

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

$$75e^{0.1t}=10$$

Divide both sides by 75.

$$e^{0.1t}=\frac{2}{15}$$

Take the logarithm of both sides of the equation.

$$\log(e^{0.1t})=\log(\frac{2}{15})$$

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

$$0.1t\log(e)=\log(\frac{2}{15})$$

Divide both sides by \log(e).

$$0.1t=\frac{\log(\frac{2}{15})}{\log(e)}$$

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

$$0.1t=\log_{e}\left(\frac{2}{15}\right)$$

Multiply both sides by 10.

$$t=\frac{\ln(\frac{2}{15})}{0.1}$$