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

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

\(\,\,\,\,\,\,\log_{b}(x^2) + \log_{b}(z) – \log_{b}(y^5) \)

\(\,\,\,\,\,\,2\log_{b}(x) + \log_{b}(z) – 5\log_{b}(y) \)

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

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

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

\(\,\,\,\,\,\,\log_{5}(\sqrt[3]{z}) – \log_{5}(25) – \log_{5}(y)^5 \)

\(\,\,\,\,\,\,\log_{5}(z)^{1/3} – \log_{5}(25) – \log_{5}(y)^5 \)

\(\,\,\,\,\,\,\frac{1}{3}\log_{5}(z) – \log_{5}(25) – 5\log_{5}(y) \)

\(\,\,\,\,\,\,\frac{1}{3}\log_{5}(z) – 2 – 5\log_{5}(y) \)

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

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

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\(\,\,\,\,\,\,\log_{2}\left(\frac{b\sqrt{y}}{32}\right) \)

\(\,\,\,\,\,\,\log_{2}(b) + \log_{2}(\sqrt{y}) – \log_{2}(32) \)

\(\,\,\,\,\,\,\log_{2}(b) + \log_{2}(y)^{1/2} – \log_{2}(32) \)

\(\,\,\,\,\,\,\log_{2}(b) + \frac{1}{2}\log_{2}(y) – \log_{2}(2^5) \)

\(\,\,\,\,\,\,\log_{2}(b) + \frac{1}{2}\log_{2}(y) – 5 \)

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

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

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

\(\,\,\,\,\,\,\log_{5}\left(4\right) + \log_{5}\left(x+3\right)^{1/2} – \log_{5}\left(x\right)\)

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

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

\(\textbf{5)}\) \( \displaystyle\log_{2}\left(\frac{x^3\sqrt{y}}{4z}\right) \)

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The answer is \( \displaystyle3\log_{2}(x)+\frac{1}{2}\log_{2}(y)-\log_{2}(4)-\log_{2}(z) \)

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\(\,\,\,\,\,\,\displaystyle\log_{2}\left(\frac{x^3\sqrt{y}}{4z}\right) \)

\(\,\,\,\,\,\,\displaystyle\log_{2}(x^3) + \log_{2}(\sqrt{y}) – \log_{2}(4) – \log_{2}(z)\)

\(\,\,\,\,\,\,\displaystyle3\log_{2}(x) + \frac{1}{2}\log_{2}(y) – \log_{2}(4) – \log_{2}(z)\)

\(\textbf{6)}\) \( \displaystyle\log_{3}\left(\frac{9y}{x^2}\right) \)

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The answer is \( \displaystyle2\log_{3}(3)+\log_{3}(y)-2\log_{3}(x) \)

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\(\,\,\,\,\,\,\displaystyle\log_{3}\left(\frac{9y}{x^2}\right) \)

\(\,\,\,\,\,\,\displaystyle\log_{3}(9) + \log_{3}(y) – \log_{3}(x^2)\)

\(\,\,\,\,\,\,\displaystyle2\log_{3}(3) + \log_{3}(y) – 2\log_{3}(x)\)

\(\textbf{7)}\) \( \displaystyle\log_{10}\left(x^2y^3\right) \)

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The answer is \( \displaystyle2\log_{10}(x)+3\log_{10}(y) \)

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\(\,\,\,\,\,\,\displaystyle\log_{10}\left(x^2y^3\right) \)

\(\,\,\,\,\,\,\displaystyle\log_{10}(x^2) + \log_{10}(y^3)\)

\(\,\,\,\,\,\,\displaystyle2\log_{10}(x) + 3\log_{10}(y)\)

\(\textbf{8)}\) \( \displaystyle\log_{6}\left(\frac{\sqrt{y}^5}{y^2}\right) \)

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The answer is \( \displaystyle\frac{5}{2}\log_{6}(y)-2\log_{6}(y) \)

Show Work

\(\,\,\,\,\,\,\displaystyle\log_{6}\left(\frac{\sqrt{y}^5}{y^2}\right) \)

\(\,\,\,\,\,\,\displaystyle\log_{6}(\sqrt{y}^5) – \log_{6}(y^2)\)

\(\,\,\,\,\,\,\displaystyle\frac{5}{2}\log_{6}(x) – 2\log_{6}(y)\)

\(\textbf{9)}\) \( \displaystyle\log_{4}\left(16\sqrt{y}\right) \)

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The answer is \( \displaystyle2+\frac{1}{2}\log_{4}(y) \)

Show Work

\(\,\,\,\,\,\,\displaystyle\log_{4}\left(16\sqrt{y}\right) \)

\(\,\,\,\,\,\,\displaystyle\log_{4}(16) + \log_{4}(\sqrt{y})\)

\(\,\,\,\,\,\,\displaystyle2 + \frac{1}{2}\log_{4}(y)\)

\(\textbf{10)}\) \( \displaystyle\log_{8}\left(\frac{64}{x^3y}\right) \)

Show Answer

The answer is \( \displaystyle2-3\log_{8}(x)-\log_{8}(y) \)

Show Work

\(\,\,\,\,\,\,\displaystyle\log_{8}\left(\frac{64}{x^3y}\right) \)

\(\,\,\,\,\,\,\displaystyle\log_{8}(64) – \log_{8}(x^3) – \log_{8}(y)\)

\(\,\,\,\,\,\,\displaystyle2 – 3\log_{8}(x) – \log_{8}(y)\)

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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