Solve 4^2x-3=4^2*2^3*4

$$4 ^ { 2 x - 3 } = 4 ^ { 2 } \times 2 ^ { 3 } \times 4$$

Solve for x

$$x = \frac{15}{4} = 3\frac{3}{4} = 3.75$$

Steps Using Definition of Logarithm

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

$$4^{2x-3}=512$$

Take the logarithm of both sides of the equation.

$$\log(4^{2x-3})=\log(512)$$

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

$$\left(2x-3\right)\log(4)=\log(512)$$

Divide both sides by \log(4).

$$2x-3=\frac{\log(512)}{\log(4)}$$

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

$$2x-3=\log_{4}\left(512\right)$$

Add 3 to both sides of the equation.

$$2x=\frac{9}{2}-\left(-3\right)$$

Divide both sides by 2.

$$x=\frac{\frac{15}{2}}{2}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{2\ln(2)}+\frac{15}{4}$$

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