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4. Logarithms of Products Logarithms of Powers Logarithms of Quotients Solving General Problems with Logarithms and Exponents Simplifying Expressions of the Form log_a a^x and a(log_a x) Properties of Logarithmic Functions Exponents and Logarithms > Properties of Logarithmic Functions Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_section_38&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dir ect&utm_source=boundless

5. The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. The logarithm of a product is the sum of the logarithms of the factors. In addition, the sum of the logarithms of two number is equal to the logarithm of the product of those two numbers. The product rule does not apply when the base of the two logarithms are different. Logarithms of Products Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/properties-of-logarithmic-functions-38/logarithms- of-products-181- 5521?campaign_content=book_196_section_38&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Exponents and Logarithms > Properties of Logarithmic Functions

6. The logarithm of a product is the sum of the logarithms of the factors. An exponent, p, signifies that a number is being multiplied by itself p number of times. Because the logarithm of a product is the sum of the logarithms of the factors, the logarithm of a number, a, to an exponent, p, is the same as the logarithm of a added together p times. The logarithm of a added together p times is the same as plogba, where b is an arbitrary base. Logarithms of Powers Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/properties-of-logarithmic-functions-38/logarithms- of-powers-182- 5902?campaign_content=book_196_section_38&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Exponents and Logarithms > Properties of Logarithmic Functions

7. The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. A basic idea in logarithmic math is that the logarithm of a product is the sum of the logarithms of the factors. A similar idea of the law of products is that the logarithm of the ratio or quotient of two numbers is the difference of the logarithms. Logarithms of Quotients Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/properties-of-logarithmic-functions-38/logarithms- of-quotients-183- 11094?campaign_content=book_196_section_38&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boun dless View on Boundless.com Exponents and Logarithms > Properties of Logarithmic Functions

8. The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. Both root equations and logarithm equations can be rewritten as exponent equations. Logarithms are therefore useful in solving equations that require solving for an exponential term, such as those involving population growth. Solving General Problems with Logarithms and Exponents Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/properties-of-logarithmic-functions-38/solving- general-problems-with-logarithms-and-exponents-184- 4643?campaign_content=book_196_section_38&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Population Growth in Sri Lanka View on Boundless.com Exponents and Logarithms > Properties of Logarithmic Functions

9. Because logaa = 1 and logabx = xlogab, the formula for the logarithm of a power says that for any number x, logaax = xlogaa = x. Because logax and logxa are inverse values, aloga(x) = x. Simplifying complex-looking equations can greatly facilitate the solving of longer problems. Simplifying Expressions of the Form log_a a^x and a(log_a x) Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/exponents-and-logarithms-5/properties-of-logarithmic-functions-38/simplifying- expressions-of-the-form-log_a-a-x-and-a-log_a-x-185- 5879?campaign_content=book_196_section_38&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com Exponents and Logarithms > Properties of Logarithmic Functions

10. Free to share, print, make copies and changes. Get yours at www.boundless.com Appendix

11. Key terms base A number raised to the power of an exponent. exponent The power to which a number, symbol, or expression is to be raised. For example, the 3 in x^3. exponential Any function that has an exponent as an independent variable. logarithm The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms

12. Interactive Graph: Graph of Binary Logarithm The graph of the logarithm with base 2 crosses the x axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3). For example, log2(8) = 3, because 23 = 8. The graph gets arbitrarily close to the y axis, but does not meet or intersect it. Try changing the value of x. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Graph of Binary Logarithm." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-graph-of-binary-logarithm- ee7f7479-dd16-483b-8ca0-771000e2c9e5 View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-graph-of-binary-logarithm- ee7f7479-dd16-483b-8ca0-771000e2c9e5View on Boundless.com Exponents and Logarithms

13. Interactive Graph: Graph of Binary Logarithm Graph of log base 2. What happens when you try Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph:." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-eaab5e4f-3692-41fd-8fee-44b052a7aadf View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-eaab5e4f-3692-41fd-8fee-44b052a7aadfView on Boundless.com Exponents and Logarithms

14. Interactive Graph: Logarithm of Powers Graph of a common logarithm,, demonstrates the rule of the logarithm power. Notice what happens when you move the slider for p. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Logarithm of Powers." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-72228e05-cb43-4c08-b77a-ab8491faaaf6 View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-72228e05-cb43-4c08-b77a-ab8491faaaf6 View on Boundless.com Exponents and Logarithms

15. Interactive Graph: Logarithms Graph of different logarithms, such as and. The graphs of logarithms of different bases have the same general shape but different curvatures. Try changing the value of x and see how it effects the curvature of the graph. Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless. "Interactive Graph: Logarithms." CC BY-SA 3.0 https://www.boundless.com/image/interactive-graph-logarithms-2c228d2d-0a9e-48cd-9062- 8ad95b80b52b View on Boundless.comCC BY-SA 3.0https://www.boundless.com/image/interactive-graph-logarithms-2c228d2d-0a9e-48cd-9062- 8ad95b80b52bView on Boundless.com Exponents and Logarithms

16. Population Growth in Sri Lanka This population growth graph shows that it grows exponentially with time. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia. "SL population growth." CC BY-SA http://commons.wikimedia.org/wiki/File:SL_population_growth.png View on Boundless.comCC BY-SAhttp://commons.wikimedia.org/wiki/File:SL_population_growth.pngView on Boundless.com Exponents and Logarithms

17. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Rewrite log(12)+log(5) using the product rule. A) log(17) B) log(12)log(5) C) log(7) D) log(60)

18. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Rewrite log(12)+log(5) using the product rule. A) log(17) B) log(12)log(5) C) log(7) D) log(60)

19. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Expand log2(xyz) using the product rule. A) (log2x)(log2y)(log2z) B) log2x+log2y+log2z C) log2x-log2y-log2z D) log(2xyz)

20. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Expand log2(xyz) using the product rule. A) (log2x)(log2y)(log2z) B) log2x+log2y+log2z C) log2x-log2y-log2z D) log(2xyz)

21. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Rewrite log3x2 as a single term using the power rule of logarithms. A) 2log3x B) 3log2x C) 6logx D) log32x

22. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Rewrite log3x2 as a single term using the power rule of logarithms. A) 2log3x B) 3log2x C) 6logx D) log32x

23. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Which of the following is equivalent to 4log2x? A) log2x4 B) log8x C) log8x D) log4x2

24. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Which of the following is equivalent to 4log2x? A) log2x4 B) log8x C) log8x D) log4x2

25. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Which of the following is equivalent to log2100-log225? A) log275 B) 4log2 C) log24 D) log42

26. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Which of the following is equivalent to log2100-log225? A) log275 B) 4log2 C) log24 D) log42

27. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Which of the following is a proper expansion of log4? A) log2-log2 B) log6-log2 C) log20-log5 D) log24

28. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Which of the following is a proper expansion of log4? A) log2-log2 B) log6-log2 C) log20-log5 D) log24

29. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Using the growth formula A=P(1+i)n, find approximately how long it will take a city that grows 6% every 2 years to double in size. A) 3 years B) 23.7913 years C) 11.8957 years D) 5.9478 years

30. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Using the growth formula A=P(1+i)n, find approximately how long it will take a city that grows 6% every 2 years to double in size. A) 3 years B) 23.7913 years C) 11.8957 years D) 5.9478 years

31. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Which of the following is equivalent to log28+log381? A) log6648 B) log589 C) 12 D) 7

32. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Which of the following is equivalent to log28+log381? A) log6648 B) log589 C) 12 D) 7

33. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms How can log22_______ be simplified? A) 18 B) 1 C) 9 D) 22

34. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms How can log22_______ be simplified? A) 18 B) 1 C) 9 D) 22

35. Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms Since we know that logaa=1, what can we infer about the value of loga1? A) It is a, because a1=a. B) It is 1, because 1a=1. C) It is 0, because a0=1. D) We don't have enough information.

36. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Exponents and Logarithms Since we know that logaa=1, what can we infer about the value of loga1? A) It is a, because a1=a. B) It is 1, because 1a=1. C) It is 0, because a0=1. D) We don't have enough information.

37. Attribution Connexions. "Logarithm Concepts -- Rewriting logarithm equations as exponent equations." CC BY 3.0 http://cnx.org/content/m18241/latest/CC BY 3.0 http://cnx.org/content/m18241/latest/ Wikipedia. "Logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/LogarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithm Connexions. "Logarithms." CC BY 3.0 http://cnx.org/content/m31883/latest/CC BY 3.0http://cnx.org/content/m31883/latest/ Wiktionary. "exponent." CC BY-SA 3.0 http://en.wiktionary.org/wiki/exponentCC BY-SA 3.0http://en.wiktionary.org/wiki/exponent Wiktionary. "exponential." CC BY-SA 3.0 http://en.wiktionary.org/wiki/exponentialCC BY-SA 3.0http://en.wiktionary.org/wiki/exponential Wikipedia. "Logarithmic function." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Logarithmic_function#Product.2C_quotient.2C_power.2C_and_rootCC BY-SA 3.0 http://en.wikipedia.org/wiki/Logarithmic_function#Product.2C_quotient.2C_power.2C_and_root Wiktionary. "exponent." CC BY-SA 3.0 http://en.wiktionary.org/wiki/exponentCC BY-SA 3.0http://en.wiktionary.org/wiki/exponent Wikipedia. "Logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Logarithm#Product.2C_quotient.2C_power.2C_and_rootCC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithm#Product.2C_quotient.2C_power.2C_and_root Wikipedia. "Logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Logarithm#Product.2C_quotient.2C_power.2C_and_rootCC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithm#Product.2C_quotient.2C_power.2C_and_root Wiktionary. "exponent." CC BY-SA 3.0 http://en.wiktionary.org/wiki/exponentCC BY-SA 3.0http://en.wiktionary.org/wiki/exponent Wikipedia. "Logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/LogarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithm Wikibooks. "Algebra/Logarithms." CC BY-SA 3.0 http://en.wikibooks.org/wiki/Algebra/LogarithmsCC BY-SA 3.0http://en.wikibooks.org/wiki/Algebra/Logarithms Wikipedia. "Logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/LogarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithm Connexions. "Logarithms." CC BY 3.0 http://cnx.org/content/m31883/latest/CC BY 3.0http://cnx.org/content/m31883/latest/ Wikipedia. "logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/logarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/logarithm Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//chemistry/definition/baseCC BY-SA 3.0http://www.boundless.com//chemistry/definition/base Wiktionary. "exponent." CC BY-SA 3.0 http://en.wiktionary.org/wiki/exponentCC BY-SA 3.0http://en.wiktionary.org/wiki/exponent Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms

38. Connexions. "Logarithm Concepts -- Properties of Logarithms." CC BY 3.0 http://cnx.org/content/m18239/latest/CC BY 3.0http://cnx.org/content/m18239/latest/ Wikipedia. "Logarithm." CC BY-SA 3.0 http://en.wikipedia.org/wiki/LogarithmCC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithm Wiktionary. "exponent." CC BY-SA 3.0 http://en.wiktionary.org/wiki/exponentCC BY-SA 3.0http://en.wiktionary.org/wiki/exponent Free to share, print, make copies and changes. Get yours at www.boundless.com Exponents and Logarithms