Solve 125^2x+1=25

$$125 ^ { 2 x + 1 } = 25$$

Solve for x

$$x=-\frac{1}{6}\approx -0.166666667$$

Steps Using Definition of Logarithm

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

$$125^{2x+1}=25$$

Take the logarithm of both sides of the equation.

$$\log(125^{2x+1})=\log(25)$$

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

$$\left(2x+1\right)\log(125)=\log(25)$$

Divide both sides by \log(125).

$$2x+1=\frac{\log(25)}{\log(125)}$$

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

$$2x+1=\log_{125}\left(25\right)$$

Subtract 1 from both sides of the equation.

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

Divide both sides by 2.

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

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{3\ln(5)}-\frac{1}{6}$$

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