Solve 27^x=1/3

$$27 ^ { x } = \frac { 1 } { 3 }$$

Solve for x

$$x=-\frac{1}{3}\approx -0.333333333$$

Steps Using Definition of Logarithm

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

$$27^{x}=\frac{1}{3}$$

Take the logarithm of both sides of the equation.

$$\log(27^{x})=\log(\frac{1}{3})$$

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

$$x\log(27)=\log(\frac{1}{3})$$

Divide both sides by \log(27).

$$x=\frac{\log(\frac{1}{3})}{\log(27)}$$

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

$$x=\log_{27}\left(\frac{1}{3}\right)$$

Solve for x (complex solution)

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

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