Solve 5^-x=4

$$5 ^ { - x } = 4$$

Solve for x

$$x=-2\log_{5}\left(2\right)\approx -0.861353116$$

Steps Using Definition of Logarithm

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

$$5^{-x}=4$$

Take the logarithm of both sides of the equation.

$$\log(5^{-x})=\log(4)$$

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

$$-x\log(5)=\log(4)$$

Divide both sides by \log(5).

$$-x=\frac{\log(4)}{\log(5)}$$

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

$$-x=\log_{5}\left(4\right)$$

Divide both sides by -1.

$$x=\frac{2\log_{5}\left(2\right)}{-1}$$

Solve for x (complex solution)

$$x=-\frac{2\pi n_{1}i}{\ln(5)}-2\log_{5}\left(2\right)$$

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