Solve 3^5x+6=1/81

$$3 ^ { 5 x + 6 } = \frac { 1 } { 81 }$$

Solve for x

$$x=-2$$

Steps Using Definition of Logarithm

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

$$3^{5x+6}=\frac{1}{81}$$

Take the logarithm of both sides of the equation.

$$\log(3^{5x+6})=\log(\frac{1}{81})$$

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

$$\left(5x+6\right)\log(3)=\log(\frac{1}{81})$$

Divide both sides by \log(3).

$$5x+6=\frac{\log(\frac{1}{81})}{\log(3)}$$

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

$$5x+6=\log_{3}\left(\frac{1}{81}\right)$$

Subtract 6 from both sides of the equation.

$$5x=-4-6$$

Divide both sides by 5.

$$x=-\frac{10}{5}$$

Solve for x (complex solution)

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

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