Calculus I - Logarithm Functions (Assignment Problems)

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Without using a calculator determine the exact value of each of the following.

\({\log _7}343\)
\({\log _4}1024\)
\(\displaystyle {\log _{\frac{3}{8}}}\frac{{27}}{{512}}\)
\(\displaystyle {\log _{11}}\frac{1}{{121}}\)
\({\log _{0.1}}0.0001\)
\({\log _{16}}4\)
\(\log 10000\)
\(\ln \frac{1}{{\sqrt[5]{{\bf{e}}}}}\)

Write each of the following in terms of simpler logarithms

\({\log _7}\left( {10{a^7}{b^3}{c^{ - 8}}} \right)\)
\(\log \left[ {{z^2}{{\left( {{x^2} + 4} \right)}^3}} \right]\)
\(\displaystyle \ln \left( {\frac{{{w^2}\,\sqrt[4]{{{t^3}}}}}{{\sqrt {t + w} }}} \right)\)

Combine each of the following into a single logarithm with a coefficient of one.

\(7\ln t - 6\ln s + 5\ln w\)
\(\displaystyle \frac{1}{2}\log \left( {z + 1} \right) - 2\log x - 4\log y - 3\log z\)
\(\displaystyle 2{\log _3}\left( {x + y} \right) + 6{\log _3}x - \frac{1}{3}\)

Use the change of base formula and a calculator to find the value of each of the following.