Solve 6^-3x=4

$$6 ^ { - 3 x } = 4$$

Solve for x

$$x=-\frac{2\log_{6}\left(2\right)}{3}\approx -0.257901871$$

Steps Using Definition of Logarithm

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

$$6^{-3x}=4$$

Take the logarithm of both sides of the equation.

$$\log(6^{-3x})=\log(4)$$

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

$$-3x\log(6)=\log(4)$$

Divide both sides by \log(6).

$$-3x=\frac{\log(4)}{\log(6)}$$

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

$$-3x=\log_{6}\left(4\right)$$

Divide both sides by -3.

$$x=\frac{2\log_{6}\left(2\right)}{-3}$$

Solve for x (complex solution)

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

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