Solve 6^-2x=6^3

$$6 ^ { - 2 x } = 6 ^ { 3 }$$

Solve for x

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

Steps Using Definition of Logarithm

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

$$6^{-2x}=216$$

Take the logarithm of both sides of the equation.

$$\log(6^{-2x})=\log(216)$$

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

$$-2x\log(6)=\log(216)$$

Divide both sides by \log(6).

$$-2x=\frac{\log(216)}{\log(6)}$$

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

$$-2x=\log_{6}\left(216\right)$$

Divide both sides by -2.

$$x=\frac{3}{-2}$$

Solve for x (complex solution)

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

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