Section 6.2 : Logarithm Functions

For problems 1 – 3 write the expression in logarithmic form.

1. \({7^5} = 16807\) Solution
2. \({16^{\frac{3}{4}}} = 8\) Solution
3. \({\left( {\displaystyle \frac{1}{3}} \right)^{ - 2}} = 9\) Solution

For problems 4 – 6 write the expression in exponential form.

1. \({\log _2}\,32 = 5\) Solution
2. \({\log _{\frac{1}{5}}}\,\displaystyle \frac{1}{{625}} = 4\) Solution
3. \({\log _9}\,\displaystyle \frac{1}{{81}} = - 2\) Solution

For problems 7 - 12 determine the exact value of each of the following without using a calculator.

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

For problems 13 – 15 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. \({\log _4}\left( {\displaystyle \frac{{x - 4}}{{{y^2}\,\sqrt[5]{z}}}} \right)\) Solution

For problems 16 – 18 combine each of the following into a single logarithm with a coefficient of one.

1. \(2{\log _4}x + 5{\log _4}y - \frac{1}{2}{\log _4}z\) 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

For problems 19 & 20 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

For problems 21 – 23 sketch each of the given functions.

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