Solve 2^a=1/3

$$2 ^ { a } = \frac { 1 } { 3 }$$

Solve for a

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

Steps Using Definition of Logarithm

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

$$2^{a}=\frac{1}{3}$$

Take the logarithm of both sides of the equation.

$$\log(2^{a})=\log(\frac{1}{3})$$

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

$$a\log(2)=\log(\frac{1}{3})$$

Divide both sides by \log(2).

$$a=\frac{\log(\frac{1}{3})}{\log(2)}$$

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

$$a=\log_{2}\left(\frac{1}{3}\right)$$

Solve for a (complex solution)

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

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