Solve 5^2x+1=1/625

$$5 ^ { 2 x + 1 } = \frac { 1 } { 625 }$$

Solve for x

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

Steps Using Definition of Logarithm

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

$$5^{2x+1}=\frac{1}{625}$$

Take the logarithm of both sides of the equation.

$$\log(5^{2x+1})=\log(\frac{1}{625})$$

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

$$\left(2x+1\right)\log(5)=\log(\frac{1}{625})$$

Divide both sides by \log(5).

$$2x+1=\frac{\log(\frac{1}{625})}{\log(5)}$$

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

$$2x+1=\log_{5}\left(\frac{1}{625}\right)$$

Subtract 1 from both sides of the equation.

$$2x=-4-1$$

Divide both sides by 2.

$$x=-\frac{5}{2}$$

Solve for x (complex solution)

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

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