Solve 3^n=2

$$3 ^ { n } = 2$$

Solve for n

$$n=\log_{3}\left(2\right)\approx 0.630929754$$

Steps Using Definition of Logarithm

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

$$3^{n}=2$$

Take the logarithm of both sides of the equation.

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

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

$$n\log(3)=\log(2)$$

Divide both sides by \log(3).

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

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

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

Solve for n (complex solution)

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

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