Solve {left(1/2right)}^3x-5=8^5

$${ \left( \frac{ 1 }{ 2 } \right) }^{ 3x-5 } = { 8 }^{ 5 }$$

Solve for x

$$x = -\frac{10}{3} = -3\frac{1}{3} \approx -3.333333333$$

Steps Using Definition of Logarithm

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

$$\left(\frac{1}{2}\right)^{3x-5}=32768$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{2}\right)^{3x-5})=\log(32768)$$

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

$$\left(3x-5\right)\log(\frac{1}{2})=\log(32768)$$

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

$$3x-5=\frac{\log(32768)}{\log(\frac{1}{2})}$$

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

$$3x-5=\log_{\frac{1}{2}}\left(32768\right)$$

Add 5 to both sides of the equation.

$$3x=-15-\left(-5\right)$$

Divide both sides by 3.

$$x=-\frac{10}{3}$$

Solve for x (complex solution)

$$x=-\frac{i\times 2\pi n_{1}}{3\ln(2)}-\frac{10}{3}$$

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