Solve 12(2^x-5)=500

$$12 ( 2 ^ { x - 5 } ) = 500$$

Solve for x

$$x=\log_{2}\left(\frac{125}{3}\right)+5\approx 10.380821784$$

Steps Using Definition of Logarithm

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

$$12\times 2^{x-5}=500$$

Divide both sides by 12.

$$2^{x-5}=\frac{125}{3}$$

Take the logarithm of both sides of the equation.

$$\log(2^{x-5})=\log(\frac{125}{3})$$

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

$$\left(x-5\right)\log(2)=\log(\frac{125}{3})$$

Divide both sides by \log(2).

$$x-5=\frac{\log(\frac{125}{3})}{\log(2)}$$

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

$$x-5=\log_{2}\left(\frac{125}{3}\right)$$

Add 5 to both sides of the equation.

$$x=\frac{\ln(\frac{125}{3})}{\ln(2)}-\left(-5\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(2)}+\log_{2}\left(\frac{125}{3}\right)+5$$

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