Evaluating Logarithms

To write the given equation in logarithmic form, we should recall the definition of a logarithm. $\begin{gathered} {\color{#FF0000}{x}}={\color{#0000FF}{b}}^{\textcolor{purple}{y}} \quad \Leftrightarrow \quad \log_{{\color{#0000FF}{b}}}{\color{#FF0000}{x}}=\textcolor{purple}{y} \end{gathered}$ The above relationship tells us that the logarithm $\textcolor{purple}{y}$ is the exponent to which ${\color{#0000FF}{b}}$ must be raised to get ${\color{#FF0000}{x}}.$ Let's now apply the definition to the given equation. $\begin{gathered} \left({\color{#0000FF}{\dfrac{1}{10}}}\right)^{\textcolor{purple}{\text{-}2}}={\color{#FF0000}{100}} \quad \Leftrightarrow \quad \log_{{\color{#0000FF}{\frac{1}{10}}}}({\color{#FF0000}{100}})=\textcolor{purple}{\text{-}2} \end{gathered}$ The above means that $\textcolor{purple}{\text{-}2}$ is the exponent to which ${\color{#0000FF}{\frac{1}{10}}}$ must be raised to obtain ${\color{#FF0000}{100}}.$