Solve 25^n=100

$$25 ^ { n } = 100$$

Solve for n

$$n=\log_{5}\left(10\right)\approx 1.430676558$$

Steps Using Definition of Logarithm

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

$$25^{n}=100$$

Take the logarithm of both sides of the equation.

$$\log(25^{n})=\log(100)$$

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

$$n\log(25)=\log(100)$$

Divide both sides by \log(25).

$$n=\frac{\log(100)}{\log(25)}$$

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

$$n=\log_{25}\left(100\right)$$

Solve for n (complex solution)

$$n=\frac{\pi n_{1}i}{\ln(5)}+\log_{5}\left(10\right)$$

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