Simplify each of the following expressions in exact form:
\log 4 + \log 9
\log_{10} \left(10\right) + \log_{10} \left(10\right)
\log_{10} 11 + \log_{10} 2 + \log_{10} 9
\log_{10} 12 - \left(\log_{10} 2 + \log_{10} 3\right)
\log_{10} 5 + \log_{10} 7 - \log_{10} 3
\dfrac{\log_{10} 4}{\log_{10} 2}
\dfrac{\log_{4} 125}{\log_{4} 5}
\dfrac{\log a^{8}}{\log a^{4}}
\dfrac{\log a^{3}}{\log \sqrt[3]{a}}
\dfrac{\log \left(\dfrac{1}{x^{4}}\right)}{\log x}
\log_{10} 10 + \dfrac{\log_{10} \left(15^{20}\right)}{\log_{10} \left(15^{5}\right)}
\dfrac{8 \log_{10} \left(\sqrt{10}\right)}{\log_{10} \left(100\right)}
10^{\log w}
\log 10 x + \log 10 y
x^{ 4 \log_{x} 3 - 6 \log_{x} 2}
Rewrite the following as the sum or difference of logarithms without any powers, fractions or surds:
\log_{9} u v
\log \left(x^{4}\right)
\log_{4} \left(x^{7}\right)
\log \left(x^{\frac{2}{5}}\right)
\log \left( 3 x^{ - 1 }\right)
\log \left(\left( 3 x\right)^{5}\right)
\log \left( 7 x^{ - 4 }\right)
\log \left(\left( 5 x\right)^{ - 7 }\right)
\log \left(\left( 2 x\right)^{ - 1 }\right)
\log \left(\dfrac{1}{x y}\right)
\log \left(\left( 3 x + 7\right)^{ - 1 }\right)
\log \left(\left( 3 x + 4\right)^{ - 8 }\right)
\log \left(\left( 5 x + 7\right)^{\frac{1}{2}}\right)
\log \left(\left( 5 x + 2\right)^{6}\right)
\log \left(\left(x + 6\right)^{5}\right)
\log_{2} \left(5x\right)
\log \left(m^{2}\right)
\log \left( 5 x^{\frac{2}{3}}\right)
\log \left(\left( 14 x\right)^{\frac{1}{3}}\right)
\log \left(\sqrt{\dfrac{c^{8}}{d}}\right)
Write each of the following as a single logarithm or integer:
5 \log x^{3} - 4 \log x^{2}
5 \log x + 3 \log y
8 \log x - \dfrac{1}{3} \log y
7 \log x - \log \left(\dfrac{1}{x}\right) - \log y
7 \log_{10} 5 - 21 \log_{10} 25
5 \log_{10} 8 - 3 \log_{10} 4
2 \log_{6} 3 + \dfrac{1}{3} \log_{6} 64
\log_{2} 36 - 2 \log_{2} 3
Rewrite the expression \log x^{2} + \log x^{3} in the form k \log x.
Rewrite the following in terms of base 10 logarithms:
\log_{4} 16
\log_{3} 0.9
\log_{3} \sqrt{5}
Rewrite \log_{3} 20 in terms of base 4 logarithms.
For each of the following logarithmic expressions:
Rewrite the expression in terms of base 10 logarithms.
Hence, evaluate each to two decimal places.
Simplify and evaluate each of the following expressions:
\dfrac{5 \log m^{2}}{6 \log \sqrt[3]{m}}
\dfrac{\log a^{8}}{\log a^{4}}
\dfrac{\log a^{3}}{\log \sqrt[3]{a}}
\dfrac{\log \left(\dfrac{1}{x^{4}}\right)}{\log x}
\dfrac{\log_{10} 4}{\log_{10} 2}
\dfrac{\log_{4} 125}{\log_{4} 5}
\log_{10} 10 + \dfrac{\log_{10} \left(15^{20}\right)}{\log_{10} \left(15^{5}\right)}
\dfrac{8 \log_{10} \left(\sqrt{10}\right)}{\log_{10} \left(100\right)}