Solve 5^2x-3+5^2=26

$${ 5 }^{ 2x-3 } + { 5 }^{ 2 } =26$$

Solve for x

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using Definition of Logarithm

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

$$5^{2x-3}+25=26$$

Subtract 25 from both sides of the equation.

$$5^{2x-3}=1$$

Take the logarithm of both sides of the equation.

$$\log(5^{2x-3})=\log(1)$$

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

$$\left(2x-3\right)\log(5)=\log(1)$$

Divide both sides by \log(5).

$$2x-3=\frac{\log(1)}{\log(5)}$$

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

$$2x-3=\log_{5}\left(1\right)$$

Add 3 to both sides of the equation.

$$2x=-\left(-3\right)$$

Divide both sides by 2.

$$x=\frac{3}{2}$$

Solve for x (complex solution)

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

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