Solve 3^x=1/6

$$3 ^ { x } = \frac { 1 } { 6 }$$

Solve for x

$$x=-\log_{3}\left(6\right)\approx -1.630929754$$

Steps Using Definition of Logarithm

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

$$3^{x}=\frac{1}{6}$$

Take the logarithm of both sides of the equation.

$$\log(3^{x})=\log(\frac{1}{6})$$

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

$$x\log(3)=\log(\frac{1}{6})$$

Divide both sides by \log(3).

$$x=\frac{\log(\frac{1}{6})}{\log(3)}$$

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

$$x=\log_{3}\left(\frac{1}{6}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(3)}-\log_{3}\left(6\right)$$

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