Solve 1/3(1/3)^x=27

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

Solve for x

$$x=-4$$

Steps Using Definition of Logarithm

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

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

Multiply both sides by 3.

$$\left(\frac{1}{3}\right)^{x}=81$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{3}\right)^{x})=\log(81)$$

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

$$x\log(\frac{1}{3})=\log(81)$$

Divide both sides by \log(\frac{1}{3}).

$$x=\frac{\log(81)}{\log(\frac{1}{3})}$$

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

$$x=\log_{\frac{1}{3}}\left(81\right)$$

Solve for x (complex solution)

$$x=-\frac{2\pi n_{1}i}{\ln(3)}-4$$

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