Solve 3^-x+5=1/27

$$3 ^ { - x + 5 } = \frac { 1 } { 27 }$$

Solve for x

$$x=8$$

Steps Using Definition of Logarithm

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

$$3^{-x+5}=\frac{1}{27}$$

Take the logarithm of both sides of the equation.

$$\log(3^{-x+5})=\log(\frac{1}{27})$$

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

$$\left(-x+5\right)\log(3)=\log(\frac{1}{27})$$

Divide both sides by \log(3).

$$-x+5=\frac{\log(\frac{1}{27})}{\log(3)}$$

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

$$-x+5=\log_{3}\left(\frac{1}{27}\right)$$

Subtract 5 from both sides of the equation.

$$-x=-3-5$$

Divide both sides by -1.

$$x=-\frac{8}{-1}$$

Solve for x (complex solution)

$$x=-\frac{i\times 2\pi n_{1}}{\ln(3)}+8$$

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