Solve 9^-x-4=27

$$9 ^ { - x - 4 } = 27$$

Solve for x

$$x = -\frac{11}{2} = -5\frac{1}{2} = -5.5$$

Steps Using Definition of Logarithm

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

$$9^{-x-4}=27$$

Take the logarithm of both sides of the equation.

$$\log(9^{-x-4})=\log(27)$$

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

$$\left(-x-4\right)\log(9)=\log(27)$$

Divide both sides by \log(9).

$$-x-4=\frac{\log(27)}{\log(9)}$$

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

$$-x-4=\log_{9}\left(27\right)$$

Add 4 to both sides of the equation.

$$-x=\frac{3}{2}-\left(-4\right)$$

Divide both sides by -1.

$$x=\frac{\frac{11}{2}}{-1}$$

Solve for x (complex solution)

$$x=-\frac{i\pi n_{1}}{\ln(3)}-\frac{11}{2}$$

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