Solve 100=325(1-0.29)^t

$$100 = 325 ( 1 - 0.29 ) ^ { t }$$

Solve for t

$$t=\log_{0.71}\left(\frac{4}{13}\right)\approx 3.441425832$$

Steps Using Definition of Logarithm

Divide both sides by 325.

$$\frac{100}{325}=\left(1-0.29\right)^{t}$$

Reduce the fraction \frac{100}{325} to lowest terms by extracting and canceling out 25.

$$\frac{4}{13}=\left(1-0.29\right)^{t}$$

Subtract 0.29 from 1 to get 0.71.

$$\frac{4}{13}=0.71^{t}$$

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

$$0.71^{t}=\frac{4}{13}$$

Take the logarithm of both sides of the equation.

$$\log(0.71^{t})=\log(\frac{4}{13})$$

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

$$t\log(0.71)=\log(\frac{4}{13})$$

Divide both sides by \log(0.71).

$$t=\frac{\log(\frac{4}{13})}{\log(0.71)}$$

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

$$t=\log_{0.71}\left(\frac{4}{13}\right)$$