Algebra - Logarithm Functions (Assignment Problems)

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For problems 1 – 5 write the expression in logarithmic form.

\(\displaystyle {11^{ - 3}} = \frac{1}{{1331}}\)
\({4^7} = 16384\)
\({\left( {\displaystyle \frac{2}{7}} \right)^{ - 3}} =\displaystyle \frac{{343}}{8}\)
\({25^{\,\frac{3}{2}}} = 125\)
\({27^{ - \,\,\frac{5}{3}}} =\displaystyle \frac{1}{{243}}\)

For problems 6 – 10 write the expression in exponential form.

\({\log _{\frac{1}{6}}}\,36 = - 2\)
\({\log _{12}}\,20736 = 4\)
\({\log _9}\,243 =\displaystyle \frac{5}{2}\)
\(\displaystyle {\log _4}\,\frac{1}{{128}} = - \frac{7}{2}\)
\({\log _8}\,32768 = 5\)

For problems 11 – 18 determine the exact value of each of the following without using a calculator.

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

For problems 19 – 20 write each of the following in terms of simpler logarithms

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

For problems 22 – 24 combine each of the following into a single logarithm with a coefficient of one.

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

For problems 25 & 26 use the change of base formula and a calculator to find the value of each of the following.

For problems 27 – 31 sketch each of the given functions.