Solve 4^2x+1=4^5

$$4 ^ { 2 x + 1 } = 4 ^ { 5 }$$

Solve for x

$$x=2$$

Steps Using Definition of Logarithm

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

$$4^{2x+1}=1024$$

Take the logarithm of both sides of the equation.

$$\log(4^{2x+1})=\log(1024)$$

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

$$\left(2x+1\right)\log(4)=\log(1024)$$

Divide both sides by \log(4).

$$2x+1=\frac{\log(1024)}{\log(4)}$$

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

$$2x+1=\log_{4}\left(1024\right)$$

Subtract 1 from both sides of the equation.

$$2x=5-1$$

Divide both sides by 2.

$$x=\frac{4}{2}$$

Solve for x (complex solution)

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

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