Functions. Essential Question What are some of the characteristics of the graph of a logarithmic function?

Transcription

1 5. Logarithms and Logarithmic Functions Essential Question What are some o the characteristics o the graph o a logarithmic unction? Ever eponential unction o the orm () = b, where b is a positive real number other than 1, has an inverse unction that ou can denote b g() = log b. This inverse unction is called a logarithmic unction with base b. Rewriting Eponential Equations Work with a partner. Find the value o in each eponential equation. Eplain our reasoning. Then use the value o to rewrite the eponential equation in its equivalent logarithmic orm, = log b. a. = 8 b. 3 = 9 c. = d. 5 = 1 e. 5 = = Graphing Eponential and Logarithmic Functions Work with a partner. Complete each table or the given eponential unction. Use the results to complete the table or the given logarithmic unction. Eplain our reasoning. Then sketch the graphs o and g in the same coordinate plane. a () = g () = log b () = 10 CONSTRUCTING VIABLE ARGUMENTS To be proicient in math, ou need to justi our conclusions and communicate them to others. g () = log Characteristics o Graphs o Logarithmic Functions Work with a partner. Use the graphs ou sketched in Eploration to determine the domain, range, -intercept, and asmptote o the graph o g() = log b, where b is a positive real number other than 1. Eplain our reasoning. Communicate Your Answer. What are some o the characteristics o the graph o a logarithmic unction? 5. How can ou use the graph o an eponential unction to obtain the graph o a logarithmic unction? Section 5. Logarithms and Logarithmic Functions 57

2 5. Lesson What You Will Learn Core Vocabular logarithm o with base b, p. 58 common logarithm, p. 59 natural logarithm, p. 59 Previous inverse unctions Deine and evaluate logarithms. Use inverse properties o logarithmic and eponential unctions. Graph logarithmic unctions. Logarithms You know that = and 3 = 8. However, or what value o does = 6? Mathematicians deine this -value using a logarithm and write = log 6. The deinition o a logarithm can be generalized as ollows. Core Concept Deinition o Logarithm with Base b Let b and be positive real numbers with b 1. The logarithm o with base b is denoted b log b and is deined as log b = i and onl i b =. The epression log b is read as log base b o. This deinition tells ou that the equations log b = and b = are equivalent. The irst is in logarithmic orm, and the second is in eponential orm. Rewriting Logarithmic Equations Rewrite each equation in eponential orm. a. log 16 = b. log 1 = 0 c. log 1 1 = 1 d. log 1/ = 1 Logarithmic Form Eponential Form a. log 16 = = 16 b. log 1 = 0 0 = 1 c. log 1 1 = = 1 d. log 1/ = 1 ( 1 ) 1 = Rewriting Eponential Equations Rewrite each equation in logarithmic orm. a. 5 = 5 b = 0.1 c. 8 /3 = d. 6 3 = 1 16 Eponential Form Logarithmic Form a. 5 = 5 log 5 5 = b = 0.1 log = 1 c. 8 /3 = log 8 = 3 d. 6 3 = 1 16 log = 3 58 Chapter 5 Eponential and Logarithmic Functions

3 Parts (b) and (c) o Eample 1 illustrate two special logarithm values that ou should learn to recognize. Let b be a positive real number such that b 1. Logarithm o 1 Logarithm o b with Base b log b 1 = 0 because b 0 = 1. log b b = 1 because b 1 = b. Evaluating Logarithmic Epressions Evaluate each logarithm. a. log 6 b. log 5 0. c. log 1/5 15 d. log 36 6 To help ou ind the value o log b, ask oursel what power o b gives ou. a. What power o gives ou 6? 3 = 6, so log 6 = 3. b. What power o 5 gives ou 0.? 5 1 = 0., so log 5 0. = 1. c. What power o 1 5 gives ou 15? ( 1 5 ) 3 = 15, so log 1/5 15 = 3. d. What power o 36 gives ou 6? 36 1/ = 6, so log 36 6 = 1. A common logarithm is a logarithm with base 10. It is denoted b log 10 or simpl b log. A natural logarithm is a logarithm with base e. It can be denoted b log e but is usuall denoted b ln. Common Logarithm log 10 = log Natural Logarithm log e = ln Evaluating Common and Natural Logarithms Evaluate (a) log 8 and (b) ln 0.3 using a calculator. Round our answer to three decimal places. Check 10^(0.903) e^(-1.0) Most calculators have kes or evaluating common and natural logarithms. a. log b. ln Check our answers b rewriting each logarithm in eponential orm and evaluating. log(8) ln(0.3) Monitoring Progress Help in English and Spanish at BigIdeasMath.com Rewrite the equation in eponential orm. 1. log 3 81 =. log 7 7 = 1 3. log 1 1 = 0. log 1/ 3 = 5 Rewrite the equation in logarithmic orm = = = /8 = Evaluate the logarithm. I necessar, use a calculator and round our answer to three decimal places. 9. log log log 1 1. ln 0.75 Section 5. Logarithms and Logarithmic Functions 59

4 Using Inverse Properties B the deinition o a logarithm, it ollows that the logarithmic unction g() = log b is the inverse o the eponential unction () = b. This means that g( ()) = log b b = and (g()) = b log b =. In other words, eponential unctions and logarithmic unctions undo each other. Simpli (a) 10 log and (b) log 5 5. Using Inverse Properties a. 10 log = b logb = b. log 5 5 = log 5 (5 ) Epress 5 as a power with base 5. = log 5 5 Power o a Power Propert = log b b = Finding Inverse Functions Find the inverse o each unction. a. () = 6 b. = ln( + 3) a. From the deinition o logarithm, the inverse o () = 6 is g() = log 6. b. = ln( + 3) Write original unction. = ln( + 3) Switch and. e = + 3 e 3 = Write in eponential orm. Subtract 3 rom each side. The inverse o = ln( + 3) is = e 3. Check a. (g()) = 6 log 6 = b. g( ()) = log 6 6 = 6 = ln( + 3) 6 = e 3 The graphs appear to be relections o each other in the line =. Monitoring Progress Simpli the epression. Help in English and Spanish at BigIdeasMath.com 60 Chapter 5 Eponential and Logarithmic Functions log 8 1. log log eln Find the inverse o =. 18. Find the inverse o = ln( 5).

5 Graphing Logarithmic Functions You can use the inverse relationship between eponential and logarithmic unctions to graph logarithmic unctions. Core Concept Parent Graphs or Logarithmic Functions The graph o () = log b is shown below or b > 1 and or 0 < b < 1. Because () = log b and g() = b are inverse unctions, the graph o () = log b is the relection o the graph o g() = b in the line =. Graph o () = log b or b > 1 Graph o () = log b or 0 < b < 1 g() = b (0, 1) (1, 0) g() = b (0, 1) (1, 0) () = log b () = log b Note that the -ais is a vertical asmptote o the graph o () = log b. The domain o () = log b is > 0, and the range is all real numbers. Graph () = log 3. Graphing a Logarithmic Function Step 1 Find the inverse o. From the deinition o logarithm, the inverse o () = log 3 is g() = 3. Step Make a table o values or g() = g() g() = 3 Step 3 Plot the points rom the table and connect them with a smooth curve. 6 Step Because () = log 3 and g() = 3 are inverse unctions, the graph o is obtained b relecting the graph o g in the line =. To do this, reverse the coordinates o the points on g and plot these new points on the graph o. 6 8 () = log 3 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the unction. 19. = log 0. () = log 5 1. = log 1/ Section 5. Logarithms and Logarithmic Functions 61

6 5. Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE A logarithm with base 10 is called a(n) logarithm.. COMPLETE THE SENTENCE The epression log 3 9 is read as. 3. WRITING Describe the relationship between = 7 and = log 7.. DIFFERENT WORDS, SAME QUESTION Which is dierent? Find both answers. What power o gives ou 16? What is log base o 16? Evaluate. Evaluate log 16. Monitoring Progress and Modeling with Mathematics In Eercises 5 10, rewrite the equation in eponential orm. (See Eample 1.) 5. log 3 9 = 6. log = 1 7. log 6 1 = 0 8. log 7 33 = 3 9. log 1/ 16 = 10. log = 1 In Eercises 11 16, rewrite the equation in logarithmic orm. (See Eample.) = = = = /3 = / = 7 In Eercises 17, evaluate the logarithm. (See Eample 3.) 17. log log log log 1/ 1 1. log log log 0.5. log NUMBER SENSE Order the logarithms rom least value to greatest value. log 5 3 log 6 38 log 7 8 log WRITING Eplain wh the epressions log ( 1) and log 1 1 are not deined. In Eercises 7 3, evaluate the logarithm using a calculator. Round our answer to three decimal places. (See Eample.) 7. log 6 8. ln 1 9. ln log ln log MODELING WITH MATHEMATICS Skdivers use an instrument called an altimeter to track their altitude as the all. The altimeter determines altitude b measuring air pressure. The altitude h (in meters) above sea level is related to the air pressure P (in pascals) b the unction shown in the diagram. What is the altitude above sea level when the air pressure is 57,000 pascals? P h = 8005 ln 101,300 h = 355 m P = 65,000 Pa h =? P = 57,000 Pa h = 738 m P = 0,000 Pa Not drawn to scale 3. MODELING WITH MATHEMATICS The ph value or a substance measures how acidic or alkaline the substance is. It is given b the ormula ph = log[h + ], where H + is the hdrogen ion concentration (in moles per liter). Find the ph o each substance. a. baking soda: [H + ] = 10 8 moles per liter b. vinegar: [H + ] = 10 3 moles per liter 6 Chapter 5 Eponential and Logarithmic Functions

7 In Eercises 35 0, simpli the epression. (See Eample 5.) log log e ln log log ln e ERROR ANALYSIS Describe and correct the error in rewriting 3 = 1 in logarithmic orm. 6 log ( 3) = 1 6. ERROR ANALYSIS Describe and correct the error in simpliing the epression log 6. log 6 = log (16 ) = log ( ) = log + = + In Eercises 3 5, ind the inverse o the unction. (See Eample 6.) 3. = 0.3. = = log 6. = log 1/5 7. = ln( 1) 8. = ln 9. = e = e 51. = = 13 + log 53. PROBLEM SOLVING The wind speed s (in miles per hour) near the center o a tornado can be modeled b s = 93 log d + 65, where d is the distance (in miles) that the tornado travels. a. In 195, a tornado traveled 0 miles through three states. Estimate the wind speed near the center o the tornado. b. Find the inverse o the given unction. Describe what the inverse represents. 5. MODELING WITH MATHEMATICS The energ magnitude M o an earthquake can be modeled b M = log E 9.9, where E is the amount o energ 3 released (in ergs). Paciic tectonic plate ault line Japan s island Honshu Eurasian tectonic plate a. In 011, a powerul earthquake in Japan, caused b the slippage o two tectonic plates along a ault, released ergs. What was the energ magnitude o the earthquake? b. Find the inverse o the given unction. Describe what the inverse represents. In Eercises 55 60, graph the unction. (See Eample 7.) 55. = log 56. = log = log 1/3 58. = log 1/ 59. = log = log 3 ( + ) USING TOOLS In Eercises 61 6, use a graphing calculator to graph the unction. Determine the domain, range, and asmptote o the unction. 61. = log( + ) 6. = ln 63. = ln( ) 6. = 3 log 65. MAKING AN ARGUMENT Your riend states that ever logarithmic unction o the orm = log b will pass through the point (1, 0). Is our riend correct? Eplain our reasoning. 66. ANALYZING RELATIONSHIPS Rank the unctions in order rom the least average rate o change to the greatest average rate o change over the interval a. = log 6 b. = log 3/5 c. 8 d. 8 8 g Section 5. Logarithms and Logarithmic Functions 63

8 67. PROBLEM SOLVING Biologists have ound that the length (in inches) o an alligator and its weight w (in pounds) are related b the unction = 7.1 ln w PROBLEM SOLVING A stud in Florida ound that the number s o ish species in a pool or lake can be modeled b the unction s = log A + 3.8(log A) where A is the area (in square meters) o the pool or lake. a. Use a graphing calculator to graph the unction. b. Use our graph to estimate the weight o an alligator that is 10 eet long. c. Use the zero eature to ind the -intercept o the graph o the unction. Does this -value make sense in the contet o the situation? Eplain. 68. HOW DO YOU SEE IT? The igure shows the graphs o the two unctions and g. 6 a. Compare the end behavior o the logarithmic unction g to that o the eponential unction. b. Determine whether the unctions are inverse unctions. Eplain. c. What is the base o each unction? Eplain. g a. Use a graphing calculator to graph the unction on the domain 00 A 35,000. b. Use our graph to estimate the number o species in a lake with an area o 30,000 square meters. c. Use our graph to estimate the area o a lake that contains si species o ish. d. Describe what happens to the number o ish species as the area o a pool or lake increases. Eplain wh our answer makes sense. 70. THOUGHT PROVOKING Write a logarithmic unction that has an output o. Then sketch the graph o our unction. 71. CRITICAL THINKING Evaluate each logarithm. (Hint: For each logarithm log b, rewrite b and as powers o the same base.) a. log 15 5 b. log 8 3 c. log 7 81 d. log 18 Maintaining Mathematical Proicienc Let () = 3. Write a rule or g that represents the indicated transormation o the graph o. (Section.3) 7. g() = () 73. g() = ( 1 ) 7. g() = ( ) g() = ( + ) Reviewing what ou learned in previous grades and lessons Identi the unction amil to which belongs. Compare the graph o to the graph o its parent unction. (Section.1) Chapter 5 Eponential and Logarithmic Functions