5.05 Logarithm laws
Rewrite the following as the sum or difference of logarithms without any powers or surds:
\log_{8} x y
\log \left(\dfrac{1}{x y}\right)
\log \left(\left( 7 x\right)^{6}\right)
\log \left( 7 x^{4}\right)
\log \left(\left( 2 x\right)^{ - 1 }\right)
\log \left(\dfrac{7}{2}\right)
\log \left( 5 x^{ - 1 }\right)
\log \left(\left( 3 x\right)^{ - 7 }\right)
\log \left( 7 x^{ - 6 }\right)
\log \left( 5 x^{\frac{2}{3}}\right)
\log \left(\left( 14 x\right)^{\frac{2}{3}}\right)
\log \left(\sqrt{\frac{c^{4}}{d}}\right)
\log \left(v^{2}\right)
Rewrite the following expressions without any powers or surds:
\log_{4} y^{7}
\log_{8} \left(x^{6}\right)
\log \left(x^{4}\right)
\log \left(\left(x + 7\right)^{6}\right)
\log \left(\left( 2 x + 5\right)^{6}\right)
\log_{a} B^{ - 5 }
\log \left(\left( 2 x + 3\right)^{ - 1 }\right)
\log \left(\left( 4 x + 3\right)^{ - 8 }\right)
\log \left(x^{\frac{1}{5}}\right)
\log_{5} \sqrt{y}
\log \left(\left( 5 x + 9\right)^{\frac{1}{3}}\right)
Write each of the following expressions as a single logarithmic term:
\log 7 + \log 12
\log_{10} 11 + \log_{10} 5 + \log_{10} 3
\log_{10} 12 - \left(\log_{10} 2 + \log_{10} 3\right)
\log_{10} 42 - \log_{10} 7
\log_{10} 8 - \log_{10} 32
\log_{10} 2 + \log_{10} 3 - \log_{10} 7
5 \left(\log_{10} 3 + \log_{10} 6\right)
3 \left(\log_{10} 6 - \log_{10} 3\right)
3 \log_{10} 22 - 3 \log_{10} 11
7 \log_{10} 5 - 21 \log_{10} 25
Write each of the following expressions as a single logarithmic term:
3 \log x^{5} - 2 \log x^{4}
6 \log x + 5 \log y
8 \log x - \dfrac{1}{2} \log y
5 \log x - \log \left(\dfrac{1}{x}\right) - \log y
\log x^{4} + \log x^{2}
\log 2 x + \log 50 y
Rewrite each of the following equations in index notation:
\log_{10} 1000 = 3
\log_{10} \left(\sqrt{10}\right) = \dfrac{1}{2}
\log_{10} \left(\dfrac{1}{1000}\right) = - 3
\log_{10} \left(\dfrac{1}{\sqrt{10}}\right) = -\dfrac{1}{2}
Rewrite each of the following equations in logarithmic notation:
10^{4} = 10\,000
10^{ - 3 } = \dfrac{1}{1000}
10^{\frac{1}{2}} = \sqrt{10}
10^{ - \frac{1}{2} } = \dfrac{1}{\sqrt{10}}
Simplify each of the following expressions:
\log_{10} 10 + \dfrac{\log_{10} \left(12^{8}\right)}{\log_{10} \left(12^{2}\right)}
\dfrac{8 \log_{10} \left(\sqrt{10}\right)}{\log_{10} \left(100\right)}
\log_{10} \left(10\right) + \log_{10} \left(10\right)
\dfrac{\log_{10} 125}{\log_{10} 5}
\dfrac{\log_{4} 49}{\log_{4} 7}
Simplify each of the following expressions:
\dfrac{5 \log m^{2}}{6 \log \sqrt[3]{m}}
\dfrac{\log a^{4}}{\log a^{2}}
\dfrac{\log a^{3}}{\log \sqrt[3]{a}}
\dfrac{\log \left(\dfrac{1}{x^{2}}\right)}{\log x}
Write \log \left(\dfrac{2 u}{3 v}\right) in terms of \log 2, \log u, \log 3 and \log v.
Rewrite the following in terms of \log u and \log v without any powers or surds:
\log \left( u^{3} v^{5}\right)
\log \left(\dfrac{\sqrt[3]{v}}{\sqrt{u}}\right)