Solve 5^x-6=100

$${ 5 }^{ x-6 } =100$$

Solve for x

$$x=2\left(\log_{5}\left(2\right)+4\right)\approx 8.861353116$$

Steps Using Definition of Logarithm

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

$$5^{x-6}=100$$

Take the logarithm of both sides of the equation.

$$\log(5^{x-6})=\log(100)$$

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

$$\left(x-6\right)\log(5)=\log(100)$$

Divide both sides by \log(5).

$$x-6=\frac{\log(100)}{\log(5)}$$

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

$$x-6=\log_{5}\left(100\right)$$

Add 6 to both sides of the equation.

$$x=2\log_{5}\left(10\right)-\left(-6\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(5)}+2\log_{5}\left(2\right)+8$$

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