Solve 2^4x-1-2=8

$$2 ^ { 4 x - 1 } - 2 = 8$$

Solve for x

$$x = \frac{\log_{2} {(5)} + 2}{4} \approx 1.080482024$$

Steps Using Definition of Logarithm

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

$$2^{4x-1}-2=8$$

Add 2 to both sides of the equation.

$$2^{4x-1}=10$$

Take the logarithm of both sides of the equation.

$$\log(2^{4x-1})=\log(10)$$

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

$$\left(4x-1\right)\log(2)=\log(10)$$

Divide both sides by \log(2).

$$4x-1=\frac{\log(10)}{\log(2)}$$

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

$$4x-1=\log_{2}\left(10\right)$$

Add 1 to both sides of the equation.

$$4x=\log_{2}\left(10\right)-\left(-1\right)$$

Divide both sides by 4.

$$x=\frac{\log_{2}\left(10\right)+1}{4}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{2\ln(2)}+\frac{\log_{2}\left(5\right)}{4}+\frac{1}{2}$$

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