Solve 13^2n-3^2=72

$$13 ^ { 2 n } - 3 ^ { 2 } = 72$$

Solve for n

$$n=2\log_{13}\left(3\right)\approx 0.856634682$$

Steps Using Definition of Logarithm

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

$$13^{2n}-9=72$$

Add 9 to both sides of the equation.

$$13^{2n}=81$$

Take the logarithm of both sides of the equation.

$$\log(13^{2n})=\log(81)$$

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

$$2n\log(13)=\log(81)$$

Divide both sides by \log(13).

$$2n=\frac{\log(81)}{\log(13)}$$

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

$$2n=\log_{13}\left(81\right)$$

Divide both sides by 2.

$$n=\frac{4\log_{13}\left(3\right)}{2}$$