Solve 4^-3x=0.1

$$4 ^ { - 3 x } = 0.1$$

Solve for x

$$x=\frac{\log_{2}\left(10\right)}{6}\approx 0.553654682$$

Steps Using Definition of Logarithm

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

$$4^{-3x}=0.1$$

Take the logarithm of both sides of the equation.

$$\log(4^{-3x})=\log(0.1)$$

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

$$-3x\log(4)=\log(0.1)$$

Divide both sides by \log(4).

$$-3x=\frac{\log(0.1)}{\log(4)}$$

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

$$-3x=\log_{4}\left(0.1\right)$$

Divide both sides by -3.

$$x=-\frac{\frac{\log_{2}\left(10\right)}{2}}{-3}$$

Solve for x (complex solution)

$$x=-\frac{i\pi n_{1}}{3\ln(2)}+\frac{\log_{2}\left(10\right)}{6}$$

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