Section 6.2 : Logarithm Functions

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

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

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

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

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

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

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

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

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

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

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

1. \({\log _7}100\)
2. \({\log _{\frac{5}{7}}}\displaystyle \frac{1}{8}\)

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

1. \(g\left( x \right) = \ln \left( { - x} \right)\)
2. \(g\left( x \right) = \ln \left( {x - 3} \right)\)
3. \(g\left( x \right) = \ln \left( x \right) + 7\)
4. \(g\left( x \right) = \ln \left( {x + 2} \right) - 4\)
5. \(g\left( x \right) = \ln \left( {x - 6} \right) + 2\)