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Notes

Practice Problems

\(\textbf{1)}\) Write as a single logarithmic expression.
\(2\log_{5}(2)+\frac{1}{2}\log_{5}(x+3)-\log_{5}(x) \)

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The answer is \( \log_{5}\left(\frac{4\sqrt{x+3}}{x}\right) \)

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\(\,\,\,\,\,\,2\log_{5}(2)+\frac{1}{2}\log_{5}(x+3)-\log_{5}(x) \)

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

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

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

\(\textbf{2)}\) Write as a single logarithmic expression.
\(2\log_{b}(x)+\log_{b}(z)-5\log_{b}(y) \)

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The answer is \( \displaystyle\log_{b}\left(\frac{x^2z}{y^5}\right) \)

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

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

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

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

\(\textbf{3)}\) Write as a single logarithmic expression.
\(\frac{1}{3}\log_{5}(z)-5\log_{5}(y)-2 \)

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The answer is \( \displaystyle\log_{5}\left(\frac{\sqrt[3]{z}}{25y^5}\right) \)

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

\(\,\,\,\,\,\,\log_{5}\left(z^{\frac{1}{3}}\right) – \log_{5}(y^5) – \log_{5}(25)\,\,\,\,\, \left(\text{note: } \log_{5}(25)=2\right) \)

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

\(\textbf{4)}\) Write as a single logarithmic expression.
\(\log_{2}(b)+\frac{1}{2}\log_{2}(z)-5 \)

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The answer is \( \displaystyle\log_{2}\left(\frac{b\sqrt{z}}{32}\right) \)

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\(\,\,\,\,\,\,\log_{2}(b)+\frac{1}{2}\log_{2}(z)-5 \)

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

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

\(\,\,\,\,\,\,\log_{2}\left(b\sqrt{z}\right) – \log_{2}(32) \)

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

\(\textbf{5)}\) Write as a single logarithmic expression.
\(2\log_{5}(x)+5\log_{5}(2)-\frac{1}{2}\log_{5}(z) \)

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The answer is \( \displaystyle\log_{5}\left(\frac{32x^2}{\sqrt{z}}\right) \)

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\(\,\,\,\,\,\,2\log_{5}(x)+5\log_{5}(2)-\frac{1}{2}\log_{5}(z) \)

\(\,\,\,\,\,\,\log_{5}(x^2) + \log_{5}(2^5) – \log_{5}(z^{\frac{1}{2}}) \)

\(\,\,\,\,\,\,\log_{5}(x^2) + \log_{5}(32) – \log_{5}(\sqrt{z}) \)

\(\,\,\,\,\,\,\log_{5}(32x^2) – \log_{5}(\sqrt{z}) \)

\(\,\,\,\,\,\,\log_{5}\left(\frac{32x^2}{\sqrt{z}}\right) \)

\(\textbf{6)}\) Write as a single logarithmic expression.
\(5\ln(x+2)-3\ln(y)-2\ln(z) \)

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The answer is \( \displaystyle\ln\left(\frac{\left(x+2\right)^5}{y^3 z^2}\right) \)

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\(\,\,\,\,\,\,5\ln(x+2)-3\ln(y)-2\ln(z) \)

\(\,\,\,\,\,\,\ln((x+2)^5) – \ln(y^3) – \ln(z^2) \)

\(\,\,\,\,\,\,\ln\left(\frac{\left(x+2\right)^5}{y^3z^2}\right) \)

\(\textbf{7)}\) Write as a single logarithmic expression.
\(\frac{1}{4}\log(x)-8\log(z)+1 \)

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The answer is \( \displaystyle\log\left(\frac{10\sqrt[4]{x}}{z^8}\right) \)

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\(\,\,\,\,\,\,\frac{1}{4}\log(x)-8\log(z)+1 \)

\(\,\,\,\,\,\,\log(x^{\frac{1}{4}}) – \log(z^8) + \log(10)\,\,\,\,\, \left(\text{note: } \log(10)=1\right)\)

\(\,\,\,\,\,\,\log(\sqrt[4]{x}) – \log(z^8) + \log(10) \)

\(\,\,\,\,\,\,\log\left(\frac{\sqrt[4]{x}}{z^8}\right) + \log(10) \)

\(\,\,\,\,\,\,\log\left(\frac{10\sqrt[4]{x}}{z^8}\right) \)

\(\textbf{8)}\) Simplify.
\(\log(8)+2\log(5)-\log(2) \)

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The answer is \( 2 \)

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\(\,\,\,\,\,\,\log(8)+2\log(5)-\log(2) \)

\(\,\,\,\,\,\,\log(2^3) + \log(5^2) – \log(2) \)

\(\,\,\,\,\,\,3\log(2) + 2\log(5) – \log(2) \)

\(\,\,\,\,\,\,\log(2^{3-1}) + \log(25) \)

\(\,\,\,\,\,\,\log(4) + \log(25) \)

\(\,\,\,\,\,\,\log(4 \cdot 25) \)

\(\,\,\,\,\,\,\log(100) \)

\(\,\,\,\,\,\,2 \)

See Related Pages\(\)

\(\bullet\text{ Expanding Logarithms}\)
\(\,\,\,\,\,\,\,\,2\log_{b}(x)+\log_{b}(z)-5\log_{b}(y)…\)

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

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

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

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

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

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

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