Solve 4^-x=37

$$4 ^ { - x } = 37$$

Solve for x

$$x = -\frac{\log_{2} {(37)}}{2} \approx -2.604726683$$

Steps Using Definition of Logarithm

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

$$4^{-x}=37$$

Take the logarithm of both sides of the equation.

$$\log(4^{-x})=\log(37)$$

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

$$-x\log(4)=\log(37)$$

Divide both sides by \log(4).

$$-x=\frac{\log(37)}{\log(4)}$$

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

$$-x=\log_{4}\left(37\right)$$

Divide both sides by -1.

$$x=\frac{\log_{2}\left(37\right)}{-2}$$

Solve for x (complex solution)

$$x=-\frac{\pi n_{1}i}{\ln(2)}-\frac{\log_{2}\left(37\right)}{2}$$

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