Solve 12^x-9-7=23

$$12 ^ { x - 9 } - 7 = 23$$

Solve for x

$$x=\log_{12}\left(154793410560\right)\approx 10.368742517$$

Steps Using Definition of Logarithm

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

$$12^{x-9}-7=23$$

Add 7 to both sides of the equation.

$$12^{x-9}=30$$

Take the logarithm of both sides of the equation.

$$\log(12^{x-9})=\log(30)$$

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

$$\left(x-9\right)\log(12)=\log(30)$$

Divide both sides by \log(12).

$$x-9=\frac{\log(30)}{\log(12)}$$

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

$$x-9=\log_{12}\left(30\right)$$

Add 9 to both sides of the equation.

$$x=\log_{12}\left(30\right)-\left(-9\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(12)}+\log_{12}\left(154793410560\right)$$

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