Solve 5^2x-3=4

$$5 ^ { 2 x - 3 } = 4$$

Solve for x

$$x=\log_{5}\left(2\right)+\frac{3}{2}\approx 1.930676558$$

Steps Using Definition of Logarithm

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

$$5^{2x-3}=4$$

Take the logarithm of both sides of the equation.

$$\log(5^{2x-3})=\log(4)$$

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

$$\left(2x-3\right)\log(5)=\log(4)$$

Divide both sides by \log(5).

$$2x-3=\frac{\log(4)}{\log(5)}$$

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

$$2x-3=\log_{5}\left(4\right)$$

Add 3 to both sides of the equation.

$$2x=2\log_{5}\left(2\right)-\left(-3\right)$$

Divide both sides by 2.

$$x=\frac{2\log_{5}\left(2\right)+3}{2}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(5)}+\log_{5}\left(2\right)+\frac{3}{2}$$

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