Solve quad8^x-1=4

$$\quad 8 ^ { x - 1 } = 4$$

Solve for x

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

Steps Using Definition of Logarithm

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

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

Take the logarithm of both sides of the equation.

$$\log(8^{x-1})=\log(4)$$

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

$$\left(x-1\right)\log(8)=\log(4)$$

Divide both sides by \log(8).

$$x-1=\frac{\log(4)}{\log(8)}$$

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

$$x-1=\log_{8}\left(4\right)$$

Add 1 to both sides of the equation.

$$x=\frac{2}{3}-\left(-1\right)$$

Solve for x (complex solution)

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

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