Solve 11^-4x=7

$$11 ^ { - 4 x } = 7$$

Solve for x

$$x=-\frac{\log_{11}\left(7\right)}{4}\approx -0.202876891$$

Steps Using Definition of Logarithm

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

$$11^{-4x}=7$$

Take the logarithm of both sides of the equation.

$$\log(11^{-4x})=\log(7)$$

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

$$-4x\log(11)=\log(7)$$

Divide both sides by \log(11).

$$-4x=\frac{\log(7)}{\log(11)}$$

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

$$-4x=\log_{11}\left(7\right)$$

Divide both sides by -4.

$$x=\frac{\log_{11}\left(7\right)}{-4}$$

Solve for x (complex solution)

$$x=-\frac{\pi n_{1}i}{2\ln(11)}-\frac{\log_{11}\left(7\right)}{4}$$

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