Solve 3^c=1/3

$$3 ^ { c } = 1 / 3$$

Solve for c

$$c=-1$$

Steps Using Definition of Logarithm

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

$$3^{c}=\frac{1}{3}$$

Take the logarithm of both sides of the equation.

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

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

$$c\log(3)=\log(\frac{1}{3})$$

Divide both sides by \log(3).

$$c=\frac{\log(\frac{1}{3})}{\log(3)}$$

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

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

Solve for c (complex solution)

$$c=\frac{2\pi n_{1}i}{\ln(3)}-1$$

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