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### 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. $${\left( \frac{1}{3}} \right)^{ - 2}} =$$ 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. $${\log _4}\left( \frac{{x - 4}}{{{y^2}\,\sqrt{z}}}} \right$$ 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