Solve 5^x=1/125

$$5 ^ { x } = \frac { 1 } { 125 }$$

Solve for x

$$x=-3$$

Steps Using Definition of Logarithm

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

$$5^{x}=\frac{1}{125}$$

Take the logarithm of both sides of the equation.

$$\log(5^{x})=\log(\frac{1}{125})$$

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

$$x\log(5)=\log(\frac{1}{125})$$

Divide both sides by \log(5).

$$x=\frac{\log(\frac{1}{125})}{\log(5)}$$

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

$$x=\log_{5}\left(\frac{1}{125}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(5)}-3$$

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