Solve 100=e^t-8

$$100 = e ^ { t - 8 }$$

Solve for t

$$t=2\left(\ln(10)+4\right)\approx 12.605170186$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$e^{t-8}=100$$

Take the logarithm of both sides of the equation.

$$\log(e^{t-8})=\log(100)$$

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

$$\left(t-8\right)\log(e)=\log(100)$$

Divide both sides by \log(e).

$$t-8=\frac{\log(100)}{\log(e)}$$

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

$$t-8=\log_{e}\left(100\right)$$

Add 8 to both sides of the equation.

$$t=2\ln(10)-\left(-8\right)$$