# 10.5E: Exercises

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### Practice Makes Perfect

##### Exercise $$\PageIndex{21}$$ Use the Properties of Logarithms

In the following exercises, use the properties of logarithms to evaluate.

1. $$\log _{4} 1$$
2. $$\log _{8} 8$$
1. $$\log _{12} 1$$
2. $$\ln e$$
1. $$3^{\log _{3} 6}$$
2. $$\log _{2} 2^{7}$$
1. $$5^{\log _{5} 10}$$
2. $$\log _{4} 4^{10}$$
1. $$8^{\log _{8} 7}$$
2. $$\log _{6} 6^{-2}$$
1. $$6^{\log _{6} 15}$$
2. $$\log _{8} 8^{-4}$$
1. $$10^{\log \sqrt{5}}$$
2. $$\log 10^{-2}$$
1. $$10^{\log \sqrt{3}}$$
2. $$\log 10^{-1}$$
1. $$e^{\ln 4}$$
2. $$\ln e^{2}$$
1. $$e^{\ln 3}$$
2. $$\ln e^{7}$$

2.

1. $$0$$
2. $$1$$

4.

1. $$10$$
2. $$10$$

6.

1. $$15$$
2. $$-4$$

8.

1. $$\sqrt{3}$$
2. $$-1$$

10.

1. $$3$$
2. $$7$$
##### Exercise $$\PageIndex{22}$$ Use the Properties of Logarithms

In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

1. $$\log _{4} 6 x$$
2. $$\log _{5} 8 y$$
3. $$\log _{2} 32 x y$$
4. $$\log _{3} 81 x y$$
5. $$\log 100 x$$
6. $$\log 1000 y$$

2. $$\log _{5} 8+\log _{5} y$$

4. $$4+\log _{3} x+\log _{3} y$$

6. $$3+\log y$$

##### Exercise $$\PageIndex{23}$$ Use the Properties of Logarithms

In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

1. $$\log _{3} \frac{3}{8}$$
2. $$\log _{6} \frac{5}{6}$$
3. $$\log _{4} \frac{16}{y}$$
4. $$\log _{5} \frac{125}{x}$$
5. $$\log \frac{x}{10}$$
6. $$\log \frac{10,000}{y}$$
7. $$\ln \frac{e^{3}}{3}$$
8. $$\ln \frac{e^{4}}{16}$$

2. $$\log _{6} 5-1$$

4. $$3-\log _{5} x$$

6. $$4-\log y$$

8. $$4-\ln 16$$

##### Exercise $$\PageIndex{24}$$ Use the Properties of Logarithms

In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible.

1. $$\log _{3} x^{2}$$
2. $$\log _{2} x^{5}$$
3. $$\log x^{-2}$$
4. $$\log x^{-3}$$
5. $$\log _{4} \sqrt{x}$$
6. $$\log _{5} \sqrt[3]{x}$$
7. $$\ln x^{\sqrt{3}}$$
8. $$\ln x^{\sqrt[3]{4}}$$

2. $$5\log _{2} x$$

4. $$-3 \log x$$

6. $$\frac{1}{3} \log _{5} x$$

8. $$\sqrt[3]{4} \ln x$$

##### Exercise $$\PageIndex{25}$$ Use the Properties of Logarithms

In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.

1. $$\log _{5}\left(4 x^{6} y^{4}\right)$$
2. $$\log _{2}\left(3 x^{5} y^{3}\right)$$
3. $$\log _{3}\left(\sqrt{2} x^{2}\right)$$
4. $$\log _{5}\left(\sqrt[4]{21} y^{3}\right)$$
5. $$\log _{3} \frac{x y^{2}}{z^{2}}$$
6. $$\log _{5} \frac{4 a b^{3} c^{4}}{d^{2}}$$
7. $$\log _{4} \frac{\sqrt{x}}{16 y^{4}}$$
8. $$\log _{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}}$$
9. $$\log _{2} \frac{\sqrt{2 x+y^{2}}}{z^{2}}$$
10. $$\log _{3} \frac{\sqrt{3 x+2 y^{2}}}{5 z^{2}}$$
11. $$\log _{2} \sqrt[4]{\frac{5 x^{3}}{2 y^{2} z^{4}}}$$
12. $$\log _{5} \sqrt[3]{\frac{3 x^{2}}{4 y^{3} z}}$$

2. $$\log _{2} 3+5 \log _{2} x+3 \log _{2} y$$

4. $$\frac{1}{4} \log _{5} 21+3 \log _{5} y$$

6. $$\begin{array}{l}{\log _{5} 4+\log _{5} a+3 \log _{5} b} {+4 \log _{5} c-2 \log _{5} d}\end{array}$$

8. $$\frac{2}{3} \log _{3} x-3-4 \log _{3} y$$

10. $$\frac{1}{2} \log _{3}\left(3 x+2 y^{2}\right)-\log _{3} 5-2 \log _{3} z$$

12. $$\begin{array}{l}{\frac{1}{3}\left(\log _{5} 3+2 \log _{5} x-\log _{5} 4\right.} {-3 \log _{5} y-\log _{5} z )}\end{array}$$

##### Exercise $$\PageIndex{26}$$ Use the Properties of Logarithms

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

1. $$\log _{6} 4+\log _{6} 9$$
2. $$\log 4+\log 25$$
3. $$\log _{2} 80-\log _{2} 5$$
4. $$\log _{3} 36-\log _{3} 4$$
5. $$\log _{3} 4+\log _{3}(x+1)$$
6. $$\log _{2} 5-\log _{2}(x-1)$$
7. $$\log _{7} 3+\log _{7} x-\log _{7} y$$
8. $$\log _{5} 2-\log _{5} x-\log _{5} y$$
9. $$4 \log _{2} x+6 \log _{2} y$$
10. $$6 \log _{3} x+9 \log _{3} y$$
11. $$\log _{3}\left(x^{2}-1\right)-2 \log _{3}(x-1)$$
12. $$\log \left(x^{2}+2 x+1\right)-2 \log (x+1)$$
13. $$4 \log x-2 \log y-3 \log z$$
14. $$3 \ln x+4 \ln y-2 \ln z$$
15. $$\frac{1}{3} \log x-3 \log (x+1)$$
16. $$2 \log (2 x+3)+\frac{1}{2} \log (x+1)$$

2. $$2$$

4. $$2$$

6. $$\log _{2} \frac{5}{x-1}$$

8. $$\log _{5} \frac{2}{x y}$$

10. $$\log _{3} x^{6} y^{9}$$

12. $$0$$

14. $$\ln \frac{x^{3} y^{4}}{z^{2}}$$

16. $$\log (2 x+3)^{2} \cdot \sqrt{x+1}$$

##### Exercise $$\PageIndex{27}$$ Use the Change-of-Base Formula

In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm.

1. $$\log _{3} 42$$
2. $$\log _{5} 46$$
3. $$\log _{12} 87$$
4. $$\log _{15} 93$$
5. $$\log _{\sqrt{2}} 17$$
6. $$\log _{\sqrt{3}} 21$$

2. $$2.379$$

4. $$1.674$$

6. $$5.542$$

##### Exercise $$\PageIndex{28}$$ Writing Exercises
1. Write the Product Property in your own words. Does it apply to each of the following? $$\log _{a} 5 x, \log _{a}(5+x)$$. Why or why not?
2. Write the Power Property in your own words. Does it apply to each of the following? $$\log _{a} x^{p},\left(\log _{a} x\right)^{r}$$. Why or why not?
3. Use an example to show that $$\log (a+b) \neq \log a+\log b ?$$
4. Explain how to find the value of $$\log _{7} 15$$ using your calculator.

b. On a scale of $$1−10$$, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?