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### Section 1.8 : Logarithm Functions

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

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

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. $$\displaystyle \ln \left( {\frac{{{w^2}\,\sqrt[4]{{{t^3}}}}}{{\sqrt {t + w} }}} \right)$$

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

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. $$\displaystyle 2{\log _3}\left( {x + y} \right) + 6{\log _3}x - \frac{1}{3}$$

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

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