Solve 4^2x+3=1

$$4 ^ { 2 x + 3 } = 1$$

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.

$$4^{2x+3}=1$$

Take the logarithm of both sides of the equation.

$$\log(4^{2x+3})=\log(1)$$

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

$$\left(2x+3\right)\log(4)=\log(1)$$

Divide both sides by \log(4).

$$2x+3=\frac{\log(1)}{\log(4)}$$

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

$$2x+3=\log_{4}\left(1\right)$$

Subtract 3 from both sides of the equation.

$$2x=-3$$

Divide both sides by 2.

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

Solve for x (complex solution)

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

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