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Section 1.8 : Logarithm Functions

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

1. \({\log _3}81\) Solution
2. \({\log _5}125\) Solution
3. \(\displaystyle {\log _2}\frac{1}{8}\) Solution
4. \(\displaystyle {\log _{\frac{1}{4}}}16\) Solution
5. \(\ln {{\bf{e}}^4}\) Solution
6. \(\displaystyle \log \frac{1}{{100}}\) Solution

Write each of the following in terms of simpler logarithms.

1. \(\log \left( {3{x^4}{y^{ - 7}}} \right)\) Solution
2. \(\ln \left( {x\sqrt {{y^2} + {z^2}} } \right)\) Solution
3. \(\displaystyle {\log _4}\left( {\frac{{x - 4}}{{{y^2}\,\sqrt[5]{z}}}} \right)\) Solution

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

1. \(\displaystyle 2{\log _4}x + 5{\log _4}y - \frac{1}{2}{\log _4}z\) Solution
2. \(3\ln \left( {t + 5} \right) - 4\ln t - 2\ln \left( {s - 1} \right)\) Solution
3. \(\displaystyle \frac{1}{3}\log a - 6\log b + 2\) Solution

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

1. \({\log _{12}}35\) Solution
2. \({\log _{\frac{2}{3}}}53\) Solution