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### 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. $$g\left( x \right) = \ln \left( {x - 6} \right) + 2$$