Solve 5^3x=1/5

$${ 5 }^{ 3x } = \frac{ 1 }{ 5 }$$

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.

$$5^{3x}=\frac{1}{5}$$

Take the logarithm of both sides of the equation.

$$\log(5^{3x})=\log(\frac{1}{5})$$

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

$$3x\log(5)=\log(\frac{1}{5})$$

Divide both sides by \log(5).

$$3x=\frac{\log(\frac{1}{5})}{\log(5)}$$

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

$$3x=\log_{5}\left(\frac{1}{5}\right)$$

Divide both sides by 3.

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

Solve for x (complex solution)

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

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