Solve 5^2x+1div125=25,f

$$5 ^ { 2 x + 1 } \div 125 = 25 , f$$

Solve for x

$$x=2$$

Steps Using Definition of Logarithm

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

$$\frac{1}{125}\times 5^{2x+1}=25$$

Multiply both sides by 125.

$$5^{2x+1}=3125$$

Take the logarithm of both sides of the equation.

$$\log(5^{2x+1})=\log(3125)$$

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

$$\left(2x+1\right)\log(5)=\log(3125)$$

Divide both sides by \log(5).

$$2x+1=\frac{\log(3125)}{\log(5)}$$

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

$$2x+1=\log_{5}\left(3125\right)$$

Subtract 1 from both sides of the equation.

$$2x=5-1$$

Divide both sides by 2.

$$x=\frac{4}{2}$$

Solve for x (complex solution)

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

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