Solve 0.9^n=0.1

$$0.9 ^ { n } = 0.1$$

Solve for n

$$n=-\log_{0.9}\left(10\right)\approx 21.854345327$$

Steps Using Definition of Logarithm

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

$$0.9^{n}=0.1$$

Take the logarithm of both sides of the equation.

$$\log(0.9^{n})=\log(0.1)$$

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

$$n\log(0.9)=\log(0.1)$$

Divide both sides by \log(0.9).

$$n=\frac{\log(0.1)}{\log(0.9)}$$

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

$$n=\log_{0.9}\left(0.1\right)$$