Solve 2^2x-4=250

$$2 ^ { 2 x - 4 } = 250$$

Solve for x

$$x = \frac{3 \log_{2} {(5)} + 5}{2} \approx 5.982892142$$

Steps Using Definition of Logarithm

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

$$2^{2x-4}=250$$

Take the logarithm of both sides of the equation.

$$\log(2^{2x-4})=\log(250)$$

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

$$\left(2x-4\right)\log(2)=\log(250)$$

Divide both sides by \log(2).

$$2x-4=\frac{\log(250)}{\log(2)}$$

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

$$2x-4=\log_{2}\left(250\right)$$

Add 4 to both sides of the equation.

$$2x=\log_{2}\left(250\right)-\left(-4\right)$$

Divide both sides by 2.

$$x=\frac{\log_{2}\left(250\right)+4}{2}$$

Solve for x (complex solution)

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

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