Solve 6^4x-13=216

$$6 ^ { 4 x - 13 } = 216$$

Solve for x

$$x=4$$

Steps Using Definition of Logarithm

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

$$6^{4x-13}=216$$

Take the logarithm of both sides of the equation.

$$\log(6^{4x-13})=\log(216)$$

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

$$\left(4x-13\right)\log(6)=\log(216)$$

Divide both sides by \log(6).

$$4x-13=\frac{\log(216)}{\log(6)}$$

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

$$4x-13=\log_{6}\left(216\right)$$

Add 13 to both sides of the equation.

$$4x=3-\left(-13\right)$$

Divide both sides by 4.

$$x=\frac{16}{4}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{2\ln(6)}+4$$

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