Solve 5^2x-3*5+2=0

$${ 5 }^{ 2x } -3 \times 5+2=0$$

Solve for x

$$x=\frac{\log_{5}\left(13\right)}{2}\approx 0.796846321$$

Steps Using Definition of Logarithm

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

$$5^{2x}-13=0$$

Add 13 to both sides of the equation.

$$5^{2x}=13$$

Take the logarithm of both sides of the equation.

$$\log(5^{2x})=\log(13)$$

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

$$2x\log(5)=\log(13)$$

Divide both sides by \log(5).

$$2x=\frac{\log(13)}{\log(5)}$$

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

$$2x=\log_{5}\left(13\right)$$

Divide both sides by 2.

$$x=\frac{\log_{5}\left(13\right)}{2}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(5)}+\frac{\log_{5}\left(13\right)}{2}$$

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