Solve 100^x-3=1000

$$100 ^ { x - 3 } = 1000$$

Solve for x

$$x = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

Steps Using Definition of Logarithm

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

$$100^{x-3}=1000$$

Take the logarithm of both sides of the equation.

$$\log(100^{x-3})=\log(1000)$$

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

$$\left(x-3\right)\log(100)=\log(1000)$$

Divide both sides by \log(100).

$$x-3=\frac{\log(1000)}{\log(100)}$$

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

$$x-3=\log_{100}\left(1000\right)$$

Add 3 to both sides of the equation.

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

Solve for x (complex solution)

$$x=\frac{9}{2}+\pi n_{1}i\log(e)$$

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