Solve 4*8^x-3=2

$$4 \cdot 8 ^ { x - 3 } = 2$$

Solve for x

$$x = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

Steps Using Definition of Logarithm

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

$$4\times 8^{x-3}=2$$

Divide both sides by 4.

$$8^{x-3}=\frac{1}{2}$$

Take the logarithm of both sides of the equation.

$$\log(8^{x-3})=\log(\frac{1}{2})$$

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

$$\left(x-3\right)\log(8)=\log(\frac{1}{2})$$

Divide both sides by \log(8).

$$x-3=\frac{\log(\frac{1}{2})}{\log(8)}$$

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

$$x-3=\log_{8}\left(\frac{1}{2}\right)$$

Add 3 to both sides of the equation.

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

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{3\ln(2)}+\frac{8}{3}$$

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