Notes
Practice Problems
\(\textbf{1)}\) Write as a single logarithmic expression.
\(2\log_{5}(2)+\frac{1}{2}\log_{5}(x+3)-\log_{5}(x) \)
The answer is \( \log_{5}\left(\frac{4\sqrt{x+3}}{x}\right) \)
\(\,\,\,\,\,\,2\log_{5}(2)+\frac{1}{2}\log_{5}(x+3)-\log_{5}(x) \)
\(\,\,\,\,\,\,\log_{5}(2^2) + \log_{5}((x+3)^{\frac{1}{2}}) – \log_{5}(x) \)
\(\,\,\,\,\,\,\log_{5}(4) + \log_{5}\left((x+3)^{\frac{1}{2}}\right) – \log_{5}(x) \)
\(\,\,\,\,\,\,\log_{5}\left(\frac{4\sqrt{x+3}}{x}\right) \)
\(\textbf{2)}\) Write as a single logarithmic expression.
\(2\log_{b}(x)+\log_{b}(z)-5\log_{b}(y) \)
The answer is \( \displaystyle\log_{b}\left(\frac{x^2z}{y^5}\right) \)
\(\,\,\,\,\,\,2\log_{b}(x)+\log_{b}(z)-5\log_{b}(y) \)
\(\,\,\,\,\,\,\log_{b}(x^2) + \log_{b}(z) – \log_{b}(y^5) \)
\(\,\,\,\,\,\,\log_{b}\left(x^2z\right) – \log_{b}(y^5) \)
\(\,\,\,\,\,\,\log_{b}\left(\frac{x^2z}{y^5}\right) \)
\(\textbf{3)}\) Write as a single logarithmic expression.
\(\frac{1}{3}\log_{5}(z)-5\log_{5}(y)-2 \)
The answer is \( \displaystyle\log_{5}\left(\frac{\sqrt[3]{z}}{25y^5}\right) \)
\(\,\,\,\,\,\,\frac{1}{3}\log_{5}(z)-5\log_{5}(y)-2 \)
\(\,\,\,\,\,\,\log_{5}\left(z^{\frac{1}{3}}\right) – \log_{5}(y^5) – \log_{5}(25)\,\,\,\,\, \left(\text{note: } \log_{5}(25)=2\right) \)
\(\,\,\,\,\,\,\log_{5}\left(\frac{\sqrt[3]{z}}{25y^5}\right) \)
\(\textbf{4)}\) Write as a single logarithmic expression.
\(\log_{2}(b)+\frac{1}{2}\log_{2}(z)-5 \)
The answer is \( \displaystyle\log_{2}\left(\frac{b\sqrt{z}}{32}\right) \)
\(\,\,\,\,\,\,\log_{2}(b)+\frac{1}{2}\log_{2}(z)-5 \)
\(\,\,\,\,\,\,\log_{2}(b) + \log_{2}(z^{\frac{1}{2}}) – \log_{2}(2^5) \)
\(\,\,\,\,\,\,\log_{2}(b) + \log_{2}(\sqrt{z}) – \log_{2}(2^5) \)
\(\,\,\,\,\,\,\log_{2}\left(b\sqrt{z}\right) – \log_{2}(32) \)
\(\,\,\,\,\,\,\log_{2}\left(\frac{b\sqrt{z}}{32}\right) \)
\(\textbf{5)}\) Write as a single logarithmic expression.
\(2\log_{5}(x)+5\log_{5}(2)-\frac{1}{2}\log_{5}(z) \)
The answer is \( \displaystyle\log_{5}\left(\frac{32x^2}{\sqrt{z}}\right) \)
\(\,\,\,\,\,\,2\log_{5}(x)+5\log_{5}(2)-\frac{1}{2}\log_{5}(z) \)
\(\,\,\,\,\,\,\log_{5}(x^2) + \log_{5}(2^5) – \log_{5}(z^{\frac{1}{2}}) \)
\(\,\,\,\,\,\,\log_{5}(x^2) + \log_{5}(32) – \log_{5}(\sqrt{z}) \)
\(\,\,\,\,\,\,\log_{5}(32x^2) – \log_{5}(\sqrt{z}) \)
\(\,\,\,\,\,\,\log_{5}\left(\frac{32x^2}{\sqrt{z}}\right) \)
\(\textbf{6)}\) Write as a single logarithmic expression.
\(5\ln(x+2)-3\ln(y)-2\ln(z) \)
The answer is \( \displaystyle\ln\left(\frac{\left(x+2\right)^5}{y^3 z^2}\right) \)
\(\,\,\,\,\,\,5\ln(x+2)-3\ln(y)-2\ln(z) \)
\(\,\,\,\,\,\,\ln((x+2)^5) – \ln(y^3) – \ln(z^2) \)
\(\,\,\,\,\,\,\ln\left(\frac{\left(x+2\right)^5}{y^3z^2}\right) \)
\(\textbf{7)}\) Write as a single logarithmic expression.
\(\frac{1}{4}\log(x)-8\log(z)+1 \)
The answer is \( \displaystyle\log\left(\frac{10\sqrt[4]{x}}{z^8}\right) \)
\(\,\,\,\,\,\,\frac{1}{4}\log(x)-8\log(z)+1 \)
\(\,\,\,\,\,\,\log(x^{\frac{1}{4}}) – \log(z^8) + \log(10)\,\,\,\,\, \left(\text{note: } \log(10)=1\right)\)
\(\,\,\,\,\,\,\log(\sqrt[4]{x}) – \log(z^8) + \log(10) \)
\(\,\,\,\,\,\,\log\left(\frac{\sqrt[4]{x}}{z^8}\right) + \log(10) \)
\(\,\,\,\,\,\,\log\left(\frac{10\sqrt[4]{x}}{z^8}\right) \)
\(\textbf{8)}\) Simplify.
\(\log(8)+2\log(5)-\log(2) \)
The answer is \( 2 \)
\(\,\,\,\,\,\,\log(8)+2\log(5)-\log(2) \)
\(\,\,\,\,\,\,\log(2^3) + \log(5^2) – \log(2) \)
\(\,\,\,\,\,\,3\log(2) + 2\log(5) – \log(2) \)
\(\,\,\,\,\,\,\log(2^{3-1}) + \log(25) \)
\(\,\,\,\,\,\,\log(4) + \log(25) \)
\(\,\,\,\,\,\,\log(4 \cdot 25) \)
\(\,\,\,\,\,\,\log(100) \)
\(\,\,\,\,\,\,2 \)
\(\textbf{9)}\) Write as a single logarithmic expression.
\(3\log_{7}(x)-\frac{1}{2}\log_{7}(y)+\log_{7}(14)\)
The answer is \(\displaystyle \log_{7}\left(\frac{14x^3}{\sqrt{y}}\right)\)
\(\,\,\,\,\,\,3\log_{7}(x)-\frac{1}{2}\log_{7}(y)+\log_{7}(14)\)
\(\,\,\,\,\,\,= \log_{7}(x^3)-\log_{7}(y^{1/2})+\log_{7}(14)\)
\(\,\,\,\,\,\,= \log_{7}\left(\frac{x^3\cdot 14}{\sqrt{y}}\right)\)
\(\textbf{10)}\) Write as a single logarithmic expression.
\(\frac{1}{2}\ln(a)+\frac{1}{3}\ln(b)-\ln(c)\)
The answer is \(\displaystyle \ln\left(\frac{\sqrt{a}\;b^{1/3}}{c}\right)\)
\(\,\,\,\,\,\,\frac{1}{2}\ln(a)+\frac{1}{3}\ln(b)-\ln(c)\)
\(\,\,\,\,\,\,=\ln\left(a^{\frac{1}{2}}\right)+\ln\left(b^{\frac{1}{3}}\right)-\ln(c)\)
\(\,\,\,\,\,\,=\ln\left(a^{\frac{1}{2}}b^{\frac{1}{3}}\right)-\ln(c)\)
\(\,\,\,\,\,\,=\ln\left(\frac{a^{\frac{1}{2}}b^{\frac{1}{3}}}{c}\right)\)
\(\textbf{11)}\) Write as a single logarithmic expression.
\(4\log_{3}(2)+\log_{3}(x)-2\log_{3}(y)+1\)
The answer is \(\displaystyle \log_{3}\left(\frac{48x}{y^2}\right)\)
\(\,\,\,\,\,\,4\log_{3}(2)+\log_{3}(x)-2\log_{3}(y)+1\)
\(\,\,\,\,\,\,=\log_{3}(2^4)+\log_{3}(x)-2\log_{3}(y)+\log_{3}(3)\)
\(\,\,\,\,\,\,=\log_{3}(16)+\log_{3}(x)-\log_{3}(y^2)+\log_{3}(3)\)
\(\,\,\,\,\,\,=\left[\log_{3}(16)+\log_{3}(3)\right]+\log_{3}(x)-\log_{3}(y^2)\)
\(\,\,\,\,\,\,=\log_{3}(16\cdot 3)+\log_{3}(x)-\log_{3}(y^2)\)
\(\,\,\,\,\,\,=\log_{3}(48)+\log_{3}(x)-\log_{3}(y^2)\)
\(\,\,\,\,\,\,=\log_{3}(48x)-\log_{3}(y^2)\)
\(\,\,\,\,\,\,=\log_{3}\left(\frac{48x}{y^2}\right)\)
\(\textbf{12)}\) Write as a single logarithmic expression.
\(\log_{10}(5)-\frac{1}{2}\log_{10}(x)+\frac{1}{3}\log_{10}(y)-\log_{10}(2)\)
The answer is \(\displaystyle \log_{10}\left(\frac{5\,y^{1/3}}{2\sqrt{x}}\right)\)
\(\,\,\,\,\,\,\log_{10}(5)-\frac{1}{2}\log_{10}(x)+\frac{1}{3}\log_{10}(y)-\log_{10}(2)\)
\(\,\,\,\,\,\,=\log_{10}(5)-\log_{10}(2)-\frac{1}{2}\log_{10}(x)+\frac{1}{3}\log_{10}(y)\)
\(\,\,\,\,\,\,=\log_{10}\left(\frac{5}{2}\right)-\log_{10}\left(x^{\frac{1}{2}}\right)+\log_{10}\left(y^{\frac{1}{3}}\right)\)
\(\,\,\,\,\,\,=\log_{10}\left(\frac{5}{2}\right)+\log_{10}\left(y^{\frac{1}{3}}\right)-\log_{10}\left(x^{\frac{1}{2}}\right)\)
\(\,\,\,\,\,\,=\log_{10}\left(\frac{\frac{5}{2}\,y^{\frac{1}{3}}}{x^{\frac{1}{2}}}\right)\)
\(\,\,\,\,\,\,=\log_{10}\left(\frac{5\,y^{1/3}}{2\sqrt{x}}\right)\)
\(\textbf{13)}\) Simplify.
\(2\ln(3)+\ln(4)-\ln(6)\)
The answer is \(\ln(6)\)
\(\,\,\,\,\,\,2\ln(3)+\ln(4)-\ln(6)\)
\(\,\,\,\,\,\,=\ln(3^2)+\ln(4)-\ln(6)\)
\(\,\,\,\,\,\,=\ln(9)+\ln(4)-\ln(6)\)
\(\,\,\,\,\,\,=\ln(9\cdot 4)-\ln(6)\)
\(\,\,\,\,\,\,=\ln(36)-\ln(6)\)
\(\,\,\,\,\,\,=\ln\left(\frac{36}{6}\right)\)
\(\,\,\,\,\,\,=\ln(6)\)
\(\textbf{14)}\) Write as a single logarithmic expression.
\(\frac{1}{3}\log_{2}(8)+\log_{2}(x)-\log_{2}(y^2)\)
The answer is \(\displaystyle \log_{2}\left(\frac{2x}{y^2}\right)\)
\(\,\,\,\,\,\,\frac{1}{3}\log_{2}(8)+\log_{2}(x)-\log_{2}(y^2)\)
\(\,\,\,\,\,\,=\log_{2}\left(8^{\frac{1}{3}}\right)+\log_{2}(x)-\log_{2}(y^2)\)
\(\,\,\,\,\,\,=\log_{2}(2)+\log_{2}(x)-\log_{2}(y^2)\)
\(\,\,\,\,\,\,=\log_{2}(2x)-\log_{2}(y^2)\)
\(\,\,\,\,\,\,=\log_{2}\left(\frac{2x}{y^2}\right)\)
\(\textbf{15)}\) Write as a single logarithmic expression.
\(-2\log_{5}(3)+3\log_{5}(10)-\frac{1}{2}\log_{5}(z)\)
The answer is \(\displaystyle \log_{5}\left(\frac{1000}{9\sqrt{z}}\right)\)
\(\,\,\,\,\,\,-2\log_{5}(3)+3\log_{5}(10)-\frac{1}{2}\log_{5}(z)\)
\(\,\,\,\,\,\,=\log_{5}(3^{-2})+\log_{5}(10^3)-\log_{5}(z^{\frac{1}{2}})\)
\(\,\,\,\,\,\,=\log_{5}\left(\frac{1}{9}\right)+\log_{5}(1000)-\log_{5}\left(\sqrt{z}\right)\)
\(\,\,\,\,\,\,=\left[\log_{5}(1000)+\log_{5}\left(\frac{1}{9}\right)\right]-\log_{5}\left(\sqrt{z}\right)\)
\(\,\,\,\,\,\,=\log_{5}\left(\frac{1000}{9}\right)-\log_{5}\left(\sqrt{z}\right)\)
\(\,\,\,\,\,\,=\log_{5}\left(\frac{1000}{9\sqrt{z}}\right)\)
\(\textbf{16)}\) Write as a single logarithmic expression.
\(\ln(2x)+\ln(3y)-\ln(4z)-\ln(5)\)
The answer is \(\displaystyle \ln\left(\frac{3xy}{10z}\right)\)
\(\,\,\,\,\,\,\ln(2x)+\ln(3y)-\ln(4z)-\ln(5)\)
\(\,\,\,\,\,\,=\ln(2x\cdot 3y)-\ln(4z\cdot 5)\)
\(\,\,\,\,\,\,=\ln(6xy)-\ln(20z)\)
\(\,\,\,\,\,\,=\ln\left(\frac{6xy}{20z}\right)\)
\(\,\,\,\,\,\,=\ln\left(\frac{3xy}{10z}\right)\)
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