Solve 3125=5^n

$$3125 = 5 ^ { n }$$

Solve for n

$$n=5$$

Steps Using Definition of Logarithm

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

$$5^{n}=3125$$

Take the logarithm of both sides of the equation.

$$\log(5^{n})=\log(3125)$$

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

$$n\log(5)=\log(3125)$$

Divide both sides by \log(5).

$$n=\frac{\log(3125)}{\log(5)}$$

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

$$n=\log_{5}\left(3125\right)$$

Solve for n (complex solution)

$$n=\frac{2\pi n_{1}i}{\ln(5)}+5$$

$$n_{1}\in \mathrm{Z}$$