Solve 1(1/2)^3x-3=8^3

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

Solve for x

$$x=-2$$

Steps Using Definition of Logarithm

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

$$\left(\frac{1}{2}\right)^{3x-3}=512$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{2}\right)^{3x-3})=\log(512)$$

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

$$\left(3x-3\right)\log(\frac{1}{2})=\log(512)$$

Divide both sides by \log(\frac{1}{2}).

$$3x-3=\frac{\log(512)}{\log(\frac{1}{2})}$$

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

$$3x-3=\log_{\frac{1}{2}}\left(512\right)$$

Add 3 to both sides of the equation.

$$3x=-9-\left(-3\right)$$

Divide both sides by 3.

$$x=-\frac{6}{3}$$

Solve for x (complex solution)

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

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