Solve 5^2x-3-8=20

$${ 5 }^{ 2x-3 } -8=20$$

Solve for x

$$x = \frac{\log_{5} {(28)} + 3}{2} \approx 2.535207536$$

Steps Using Definition of Logarithm

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

$$5^{2x-3}-8=20$$

Add 8 to both sides of the equation.

$$5^{2x-3}=28$$

Take the logarithm of both sides of the equation.

$$\log(5^{2x-3})=\log(28)$$

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

$$\left(2x-3\right)\log(5)=\log(28)$$

Divide both sides by \log(5).

$$2x-3=\frac{\log(28)}{\log(5)}$$

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

$$2x-3=\log_{5}\left(28\right)$$

Add 3 to both sides of the equation.

$$2x=\log_{5}\left(28\right)-\left(-3\right)$$

Divide both sides by 2.

$$x=\frac{\log_{5}\left(28\right)+3}{2}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(5)}+\frac{\log_{5}\left(28\right)}{2}+\frac{3}{2}$$

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