Solve (1/2)^frac{x{3}+1}=1/6

$$( \frac { 1 } { 2 } ) ^ { \frac { x } { 3 } + 1 } = \frac { 1 } { 6 }$$

Solve for x

$$x=3\log_{2}\left(3\right)\approx 4.754887502$$

Steps Using Definition of Logarithm

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

$$\left(\frac{1}{2}\right)^{\frac{1}{3}x+1}=\frac{1}{6}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{2}\right)^{\frac{1}{3}x+1})=\log(\frac{1}{6})$$

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

$$\left(\frac{1}{3}x+1\right)\log(\frac{1}{2})=\log(\frac{1}{6})$$

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

$$\frac{1}{3}x+1=\frac{\log(\frac{1}{6})}{\log(\frac{1}{2})}$$

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

$$\frac{1}{3}x+1=\log_{\frac{1}{2}}\left(\frac{1}{6}\right)$$

Subtract 1 from both sides of the equation.

$$\frac{1}{3}x=\log_{2}\left(6\right)-1$$

Multiply both sides by 3.

$$x=\frac{\log_{2}\left(6\right)-1}{\frac{1}{3}}$$

Solve for x (complex solution)

$$x=-\frac{6\pi n_{1}i}{\ln(2)}+3\log_{2}\left(3\right)$$

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