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Evaluating Logarithms

Evaluating Logarithms 1.1 - Solution

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To write the given equation in logarithmic form, we should recall the definition of a logarithm. x=bylogbx=y\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 y\textcolor{purple}{y} is the exponent to which b{\color{#0000FF}{b}} must be raised to get x.{\color{#FF0000}{x}}. Let's now apply the definition to the given equation. (110)-2=100log110(100)=-2\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 -2\textcolor{purple}{\text{-}2} is the exponent to which 110{\color{#0000FF}{\frac{1}{10}}} must be raised to obtain 100.{\color{#FF0000}{100}}.