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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. $$\displaystyle {\log _4}\left( {\frac{{x - 4}}{{{y^2}\,\sqrt{z}}}} \right)$$ 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