Solve 4^-x=3.7

$$4 ^ { - x } = 3.7$$

Solve for x

$$x=\frac{\log_{2}\left(\frac{10}{37}\right)}{2}\approx -0.943762635$$

Steps Using Definition of Logarithm

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

$$4^{-x}=3.7$$

Take the logarithm of both sides of the equation.

$$\log(4^{-x})=\log(3.7)$$

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

$$-x\log(4)=\log(3.7)$$

Divide both sides by \log(4).

$$-x=\frac{\log(3.7)}{\log(4)}$$

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

$$-x=\log_{4}\left(3.7\right)$$

Divide both sides by -1.

$$x=\frac{\ln(\frac{37}{10})}{-2\ln(2)}$$