Solve 81^2x+5=(1/3)^2

$$81 ^ { 2 x + 5 } = ( \frac { 1 } { 3 } ) ^ { 2 }$$

Solve for x

$$x = -\frac{11}{4} = -2\frac{3}{4} = -2.75$$

Steps Using Definition of Logarithm

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

$$81^{2x+5}=\frac{1}{9}$$

Take the logarithm of both sides of the equation.

$$\log(81^{2x+5})=\log(\frac{1}{9})$$

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

$$\left(2x+5\right)\log(81)=\log(\frac{1}{9})$$

Divide both sides by \log(81).

$$2x+5=\frac{\log(\frac{1}{9})}{\log(81)}$$

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

$$2x+5=\log_{81}\left(\frac{1}{9}\right)$$

Subtract 5 from both sides of the equation.

$$2x=-\frac{1}{2}-5$$

Divide both sides by 2.

$$x=-\frac{\frac{11}{2}}{2}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{4\ln(3)}-\frac{11}{4}$$

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