Solve (1/2)^10-3x=32

$$( \frac { 1 } { 2 } ) ^ { 10 - 3 x } = 32$$

Solve for x

$$x=5$$

Steps Using Definition of Logarithm

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

$$\left(\frac{1}{2}\right)^{-3x+10}=32$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{2}\right)^{-3x+10})=\log(32)$$

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

$$\left(-3x+10\right)\log(\frac{1}{2})=\log(32)$$

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

$$-3x+10=\frac{\log(32)}{\log(\frac{1}{2})}$$

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

$$-3x+10=\log_{\frac{1}{2}}\left(32\right)$$

Subtract 10 from both sides of the equation.

$$-3x=-5-10$$

Divide both sides by -3.

$$x=-\frac{15}{-3}$$

Solve for x (complex solution)

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

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