Solve 1/25=(5/6)^n-1

$$\frac { 1 } { 25 } = ( \frac { 5 } { 6 } ) ^ { n - 1 }$$

Solve for n

$$n=-\log_{\frac{5}{6}}\left(30\right)\approx 18.654938239$$

Steps Using Definition of Logarithm

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

$$\left(\frac{5}{6}\right)^{n-1}=\frac{1}{25}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{5}{6}\right)^{n-1})=\log(\frac{1}{25})$$

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

$$\left(n-1\right)\log(\frac{5}{6})=\log(\frac{1}{25})$$

Divide both sides by \log(\frac{5}{6}).

$$n-1=\frac{\log(\frac{1}{25})}{\log(\frac{5}{6})}$$

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

$$n-1=\log_{\frac{5}{6}}\left(\frac{1}{25}\right)$$

Add 1 to both sides of the equation.

$$n=-\frac{2\ln(5)}{\ln(\frac{5}{6})}-\left(-1\right)$$