Solve 9^n=12

$$9 ^ { n } = 12$$

Solve for n

$$n=\log_{3}\left(2\right)+\frac{1}{2}\approx 1.130929754$$

Steps Using Definition of Logarithm

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

$$9^{n}=12$$

Take the logarithm of both sides of the equation.

$$\log(9^{n})=\log(12)$$

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

$$n\log(9)=\log(12)$$

Divide both sides by \log(9).

$$n=\frac{\log(12)}{\log(9)}$$

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

$$n=\log_{9}\left(12\right)$$

Solve for n (complex solution)

$$n=\frac{\pi n_{1}i}{\ln(3)}+\frac{\log_{3}\left(12\right)}{2}$$

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