Solve 2=3^m

$$2 = 3 ^ { m }$$

Solve for m

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

Steps Using Definition of Logarithm

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

$$3^{m}=2$$

Take the logarithm of both sides of the equation.

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

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

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

Divide both sides by \log(3).

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

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

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

Solve for m (complex solution)

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

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