Solve 3=2^k

$$3 = 2 ^ { k }$$

Solve for k

$$k=\log_{2}\left(3\right)\approx 1.584962501$$

Steps Using Definition of Logarithm

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

$$2^{k}=3$$

Take the logarithm of both sides of the equation.

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

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

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

Divide both sides by \log(2).

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

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

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

Solve for k (complex solution)

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

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