Evaluate log base 0.1 of 100

The change of base rule can be used if ${\displaystyle a}$ and ${\displaystyle b}$ are greater than ${\displaystyle 0}$ and not equal to ${\displaystyle 1}$, and ${\displaystyle x}$ is greater than ${\displaystyle 0}$.

Substitute in values for the variables in the change of base formula, using ${\displaystyle b=10}$.

Rewrite as an equation.

Rewrite ${\displaystyle \log \left(100\right)=x}$ in exponential form using the definition of a logarithm. If ${\displaystyle x}$ and ${\displaystyle b}$ are positive real numbers and ${\displaystyle b}$ does not equal ${\displaystyle 1}$, then ${\displaystyle {\log }_b\left(x\right)=y}$ is equivalent to ${\displaystyle b^y=x}$.

Create equivalent expressions in the equation that all have equal bases.

Since the bases are the same, the two expressions are only equal if the exponents are also equal.

The variable ${\displaystyle x}$ is equal to ${\displaystyle 2}$.