Solve 32^2x-3=2

$$32 ^ { 2 x - 3 } = 2$$

Solve for x

$$x = \frac{8}{5} = 1\frac{3}{5} = 1.6$$

Steps Using Definition of Logarithm

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

$$32^{2x-3}=2$$

Take the logarithm of both sides of the equation.

$$\log(32^{2x-3})=\log(2)$$

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

$$\left(2x-3\right)\log(32)=\log(2)$$

Divide both sides by \log(32).

$$2x-3=\frac{\log(2)}{\log(32)}$$

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

$$2x-3=\log_{32}\left(2\right)$$

Add 3 to both sides of the equation.

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

Divide both sides by 2.

$$x=\frac{\frac{16}{5}}{2}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{5\ln(2)}+\frac{8}{5}$$

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