Solve 1/81=3^x+1

$$\frac{ 1 }{ 81 } = { 3 }^{ x+1 }$$

Solve for x

$$x=-5$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$3^{x+1}=\frac{1}{81}$$

Take the logarithm of both sides of the equation.

$$\log(3^{x+1})=\log(\frac{1}{81})$$

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

$$\left(x+1\right)\log(3)=\log(\frac{1}{81})$$

Divide both sides by \log(3).

$$x+1=\frac{\log(\frac{1}{81})}{\log(3)}$$

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

$$x+1=\log_{3}\left(\frac{1}{81}\right)$$

Subtract 1 from both sides of the equation.

$$x=-4-1$$

Solve for x (complex solution)

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

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