Solve 12^2x-8=15

$$12 ^ { 2 x - 8 } = 15$$

Solve for x

$$x = \frac{\log_{12} {(6449725440)}}{2} \approx 4.544899786$$

Steps Using Definition of Logarithm

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

$$12^{2x-8}=15$$

Take the logarithm of both sides of the equation.

$$\log(12^{2x-8})=\log(15)$$

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

$$\left(2x-8\right)\log(12)=\log(15)$$

Divide both sides by \log(12).

$$2x-8=\frac{\log(15)}{\log(12)}$$

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

$$2x-8=\log_{12}\left(15\right)$$

Add 8 to both sides of the equation.

$$2x=\log_{12}\left(15\right)-\left(-8\right)$$

Divide both sides by 2.

$$x=\frac{\log_{12}\left(15\right)+8}{2}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(12)}+\frac{\log_{12}\left(6449725440\right)}{2}$$

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