Solve 125^2n+3=625

$$125 ^ { 2 n + 3 } = 625$$

Solve for n

$$n=-\frac{5}{6}\approx -0.833333333$$

Steps Using Definition of Logarithm

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

$$125^{2n+3}=625$$

Take the logarithm of both sides of the equation.

$$\log(125^{2n+3})=\log(625)$$

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

$$\left(2n+3\right)\log(125)=\log(625)$$

Divide both sides by \log(125).

$$2n+3=\frac{\log(625)}{\log(125)}$$

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

$$2n+3=\log_{125}\left(625\right)$$

Subtract 3 from both sides of the equation.

$$2n=\frac{4}{3}-3$$

Divide both sides by 2.

$$n=-\frac{\frac{5}{3}}{2}$$