Solve 4^x-2=8^6

$$4 ^ { x - 2 } = 8 ^ { 6 }$$

Solve for x

$$x=11$$

Steps Using Definition of Logarithm

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

$$4^{x-2}=262144$$

Take the logarithm of both sides of the equation.

$$\log(4^{x-2})=\log(262144)$$

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

$$\left(x-2\right)\log(4)=\log(262144)$$

Divide both sides by \log(4).

$$x-2=\frac{\log(262144)}{\log(4)}$$

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

$$x-2=\log_{4}\left(262144\right)$$

Add 2 to both sides of the equation.

$$x=9-\left(-2\right)$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(2)}+11$$

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