Solve 25^3x+1=1/5

$$25 ^ { 3 x + 1 } = \frac { 1 } { 5 }$$

Solve for x

$$x=-\frac{1}{2}=-0.5$$

Steps Using Definition of Logarithm

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

$$25^{3x+1}=\frac{1}{5}$$

Take the logarithm of both sides of the equation.

$$\log(25^{3x+1})=\log(\frac{1}{5})$$

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

$$\left(3x+1\right)\log(25)=\log(\frac{1}{5})$$

Divide both sides by \log(25).

$$3x+1=\frac{\log(\frac{1}{5})}{\log(25)}$$

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

$$3x+1=\log_{25}\left(\frac{1}{5}\right)$$

Subtract 1 from both sides of the equation.

$$3x=-\frac{1}{2}-1$$

Divide both sides by 3.

$$x=-\frac{\frac{3}{2}}{3}$$

Solve for x (complex solution)

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

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