Solve (7)(1/3)^x=1/27

$$( 7 ) ( \frac { 1 } { 3 } ) ^ { x } = \frac { 1 } { 27 }$$

Solve for x

$$x=\log_{3}\left(189\right)\approx 4.771243749$$

Steps Using Definition of Logarithm

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

$$7\times \left(\frac{1}{3}\right)^{x}=\frac{1}{27}$$

Divide both sides by 7.

$$\left(\frac{1}{3}\right)^{x}=\frac{1}{189}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{3}\right)^{x})=\log(\frac{1}{189})$$

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

$$x\log(\frac{1}{3})=\log(\frac{1}{189})$$

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

$$x=\frac{\log(\frac{1}{189})}{\log(\frac{1}{3})}$$

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

$$x=\log_{\frac{1}{3}}\left(\frac{1}{189}\right)$$

Solve for x (complex solution)

$$x=-\frac{2\pi n_{1}i}{\ln(3)}+\log_{3}\left(189\right)$$

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