Solve 10^5=100^x

$$10 ^ { 5 } = 100 ^ { x }$$

Solve for x

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Steps Using Definition of Logarithm

Calculate 10 to the power of 5 and get 100000.

$$100000=100^{x}$$

Swap sides so that all variable terms are on the left hand side.

$$100^{x}=100000$$

Take the logarithm of both sides of the equation.

$$\log(100^{x})=\log(100000)$$

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

$$x\log(100)=\log(100000)$$

Divide both sides by \log(100).

$$x=\frac{\log(100000)}{\log(100)}$$

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

$$x=\log_{100}\left(100000\right)$$

Solve for x (complex solution)

$$x=\frac{5}{2}+\pi n_{1}i\log(e)$$

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