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

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

Solve for x

$$x=-2\log_{3}\left(2\right)-1\approx -2.261859507$$

Steps Using Definition of Logarithm

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

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

Divide both sides by 2.

$$\left(\frac{1}{3}\right)^{x+1}=4$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{3}\right)^{x+1})=\log(4)$$

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

$$\left(x+1\right)\log(\frac{1}{3})=\log(4)$$

Divide both sides by \log(\frac{1}{3}).

$$x+1=\frac{\log(4)}{\log(\frac{1}{3})}$$

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

$$x+1=\log_{\frac{1}{3}}\left(4\right)$$

Subtract 1 from both sides of the equation.

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

Solve for x (complex solution)

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

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