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Notes

Practice Problems

Expand the following logarithms.

\(\textbf{1)}\) \( \displaystyle\log_{b}\left(\frac{x^2z}{y^5}\right) \)

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The answer is \( 2\log_{b}(x)+\log_{b}(z)-5\log_{b}(y) \)

\(\textbf{2)}\) \( \displaystyle\log_{5}\left(\frac{\sqrt[3]{z}}{25y^5}\right) \)

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The answer is \( \frac{1}{3}\log_{5}(z)-5\log_{5}(y)-2 \)

\(\textbf{3)}\) \( \displaystyle\log_{2}\left(\frac{b\sqrt{n}}{32}\right) \)

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The answer is \( \log_{2}(b)+\frac{1}{2}\log_{2}(n)-5 \)

\(\textbf{4)}\) \( \displaystyle\log_{5}\left(\frac{4\sqrt{x+3}}{x}\right) \)

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The answer is \( \displaystyle2\log_{5}(2)+\frac{1}{2}\log_{5}(x+3)-\log_{5}(x) \)

Show Work

\(\,\,\,\,\,\,\displaystyle\log_{5}\left(\frac{4\sqrt{x+3}}{x}\right) \)

\(\,\,\,\,\,\,\displaystyle\log_{5}\left(4\right) + \log_{5}\left(\sqrt{x+3}\right) – \log_{5}\left(x\right)\)

\(\,\,\,\,\,\,\displaystyle\log_{5}\left(2\right)^2 + \log_{5}(x+3)^{1/2} – \log_{5}\left(x\right)\)

\(\,\,\,\,\,\,2\log_{5}\left(2\right) + \frac{1}{2} \log_{5}(x+3) – \log_{5}\left(x\right)\)

See Related Pages\(\)

\(\bullet\text{ Evaluating Logarithms}\)
\(\,\,\,\,\,\,\,\,\log_{2}(8)…\)

\(\bullet\text{ Decibel Problems}\)
\(\,\,\,\,\,\,\,\,N_{dB}=10\log \left(\frac{P}{10^{-12}}\right)…\)

\(\bullet\text{ Earthquake Problems}\)
\(\,\,\,\,\,\,\,\,M=\log\frac{I}{10^{-4}}…\)

\(\bullet\text{ Domain and Range Logarithmic Functions}\)
\(\,\,\,\,\,\,\,\,f(x)=log(x) \rightarrow \text{Domain:} x\gt0… \)

\(\bullet\text{ Graphing Logarithmic Functions}\)
\(\,\,\,\,\,\,\,\,f(x)=log_{2}(x)\)

\(\bullet\text{ Solving Logarithmic Equations}\)
\(\,\,\,\,\,\,\,\,\log_{2}(5x)=\log_{2}(2x+12)…\)

\(\bullet\text{ Inverse of Logarithmic Functions}\)
\(\,\,\,\,\,\,\,\,f(x)=log_{2}(x) \rightarrow f^{-1}(x)=2^x\)

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