Evaluate Logarithms and Graph Logarithmic Functions

Transcription

1 TEKS 7.4 2A.4.C, 2A..A, 2A..B, 2A..C Before Now Evaluate Logarithms and Graph Logarithmic Functions You evaluated and graphed eponential functions. You will evaluate logarithms and graph logarithmic functions. Wh? So ou can model the wind speed of a tornado, as in Eample 4. Ke Vocabular logarithm of with base b common logarithm natural logarithm You know that and However, for what value of does 2 5 6? Mathematicians define this -value using a logarithm and write 5 log 2 6. The definition of a logarithm can be generalized as follows. KEY CONCEPT For Your Notebook Definition of Logarithm with Base b Let b and be positive numbers with b Þ. The logarithm of with base b is denoted b log b and is defined as follows: log b 5 if and onl if b 5 The epression log b is read as log base b of. This definition tells ou that the equations log b 5 and b 5 are equivalent. The first is in logarithmic form and the second is in eponential form. E XAMPLE Rewrite logarithmic equations Logarithmic Form Eponential Form a. log b. log c. log d. log / } Parts (b) and (c) of Eample illustrate two special logarithm values that ou should learn to recognize. Let b be a positive real number such that b Þ. Logarithm of Logarithm of b with Base b log b 5 0 because b 0 5. log b b 5 because b 5 b. GUIDED PRACTICE for Eample Rewrite the equation in eponential form.. log log log log / Evaluate Logarithms and Graph Logarithmic Functions 499

2 E XAMPLE 2 Evaluate logarithms Evaluate the logarithm. a. log 4 64 b. log c. log /5 25 d. log 36 6 To help ou find the value of log b, ask ourself what power of b gives ou. a. 4 to what power gives 64? , so log b. 5 to what power gives 0.2? , so log c. } 5 to what power gives 25? } , so log / d. 36 to what power gives 6? 36 /2 5 6, so log } 2. SPECIAL LOGARITHMS A common logarithm is a logarithm with base 0. It is denoted b log 0 or simpl b log. A natural logarithm is a logarithm with base e. It can be denoted b log e, but is more often denoted b ln. Common Logarithm log 0 5 log Natural Logarithm log e 5 ln Most calculators have kes for evaluating common and natural logarithms. E XAMPLE 3 Evaluate common and natural logarithms Epression Kestrokes Displa Check a. log ø 8 b. ln e ø 0.3 E XAMPLE 4 Evaluate a logarithmic model TORNADOES The wind speed s (in miles per hour) near the center of a tornado can be modeled b s 5 93 log d 65 where d is the distance (in miles) that the tornado travels. In 925, a tornado traveled 220 miles through three states. Estimate the wind speed near the tornado s center. s 5 93 log d 65 Write function log Substitute 220 for d. ø 93(2.342) 65 Use a calculator. Not drawn to scale Simplif. c The wind speed near the tornado s center was about 283 miles per hour. 500 Chapter 7 Eponential and Logarithmic Functions

3 GUIDED PRACTICE for Eamples 2, 3, and 4 Evaluate the logarithm. Use a calculator if necessar. 5. log log log 2 8. ln WHAT IF? Use the function in Eample 4 to estimate the wind speed near a tornado s center if its path is 50 miles long. INVERSE FUNCTIONS B the definition of a logarithm, it follows that the logarithmic function g() 5 log b is the inverse of the eponential function f() 5 b. This means that: g(f()) 5 log b b 5 and f(g()) 5 b log b 5 E XAMPLE 5 Use inverse properties Simplif the epression. a. 0 log 4 b. log 5 25 a. 0 log b log b 5 b. log log 5 (5 2 ) Epress 25 as a power with base 5. 5 log Power of a power propert 5 2 log b b 5 E XAMPLE 6 Find inverse functions Find the inverse of the function. a. 5 6 b. 5 ln ( 3) REVIEW INVERSES For help with finding inverses of functions, see p a. From the definition of logarithm, the inverse of 5 6 is 5 log 6. b. 5 ln ( 3) Write original function. 5 ln ( 3) Switch and. e 5 3 Write in eponential form. e Solve for. c The inverse of 5 ln ( 3) is 5 e 2 3. GUIDED PRACTICE for Eamples 5 and 6 Simplif the epression log 8. log log 2 64 ln e 4. Find the inverse of Find the inverse of 5 ln ( 2 5). 7.4 Evaluate Logarithms and Graph Logarithmic Functions 50

4 GRAPHING LOGARITHMIC FUNCTIONS You can use the inverse relationship between eponential and logarithmic functions to graph logarithmic functions. KEY CONCEPT For Your Notebook Parent Graphs for Logarithmic Functions The graph of f() 5 log b is shown below for b > and for 0 < b <. Because f() 5 log b and g() 5 b are inverse functions, the graph of f() 5 log b is the reflection of the graph of g() 5 b in the line 5. Graph of f () 5 log b for b > Graph of f () 5 log b for 0 < b < g()5 b (0, ) (, 0) g()5 b (0, ) (, 0) f() 5 log b f()5 log b Note that the -ais is a vertical asmptote of the graph of f() 5 log b. The domain of f() 5 log b is > 0, and the range is all real numbers. E XAMPLE 7 Graph logarithmic functions Graph the function. a. 5 log 3 b. 5 log /2 a. Plot several convenient points, such as (, 0), (3, ), and (9, 2). The -ais is a vertical asmptote. From left to right, draw a curve that starts just to the right of the -ais and moves up through the plotted points, as shown below. b. Plot several convenient points, such as (, 0), (2, 2), (4, 22), and (8, 23). The -ais is a vertical asmptote. From left to right, draw a curve that starts just to the right of the -ais and moves down through the plotted points, as shown below. (3, ) (, 0) 4 (9, 2) (, 0) 3 (2, 2) (4, 22) (8, 23) at classzone.com 502 Chapter 7 Eponential and Logarithmic Functions

5 TRANSLATIONS You can graph a logarithmic function of the form 5 log b ( 2 h) k b translating the graph of the parent function 5 log b. E XAMPLE 8 Translate a logarithmic graph Graph 5 log 2 ( 3). State the domain and range. STEP Sketch the graph of the parent function 5 log 2, which passes through (, 0), (2, ), and (4, 2). STEP 2 Translate the parent graph left 3 units and up unit. The translated graph passes through (22, ), (2, 2), and (, 3). The graph s asmptote is 523. The domain is > 23, and the range is all real numbers. (2, 2) (22, ) 5 log 2 ( 3) 4 (, 3) (4, 2) (2, ) (, 0) 4 5 log 2 GUIDED PRACTICE for Eamples 7 and 8 Graph the function. State the domain and range log log /3 ( 2 3) 8. f() 5 log 4 ( ) EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 3, 33, and 6 5 TAKS PRACTICE AND REASONING Es. 36, 6, 62, 64, and 65. VOCABULARY Cop and complete: A logarithm with base 0 is called a(n)? logarithm. 2. WRITING Describe the relationship between 5 5 and 5 log 5. EXAMPLE on p. 499 for Es. 3 7 EXPONENTIAL FORM Rewrite the equation in eponential form. 3. log log log 6 } log ERROR ANALYSIS Describe and correct the error in rewriting the equation } in logarithmic form. 8 log } 8 EXAMPLE 2 on p. 500 for Es. 8 9 EVALUATING LOGARITHMS Evaluate the logarithm without using a calculator. 8. log log log log log 9 3. log / log 3 }27 5. log 6 }4 6. log / log log log Evaluate Logarithms and Graph Logarithmic Functions 503

6 EXAMPLE 3 on p. 500 for Es EXAMPLE 5 on p. 50 for Es CALCULATING LOGARITHMS Use a calculator to evaluate the logarithm. 20. log 4 2. ln ln log log ln log ln 0 USING INVERSE PROPERTIES Simplif the epression log log log log log log log log MULTIPLE TAKS REASONING CHOICE Which epression is equivalent to log 00? A B 2 C 0 D 00 EXAMPLE 6 on p. 50 for Es EXAMPLES 7 and 8 on pp for Es FINDING INVERSES Find the inverse of the function log (0.4) log / e ln ( ) log GRAPHING FUNCTIONS Graph the function. State the domain and range log log log / log / log 2 ( 2 3) log f() 5 log 4 ( 2) g() 5 log 6 ( 2 4) h() 5 log 5 ( ) 2 3 CHALLENGE Evaluate the logarithm. (Hint: For each logarithm log b, rewrite b and as powers of the same number.) 54. log log log log 4 28 PROBLEM SOLVING EXAMPLE 4 on p. 500 for Es ALTIMETER Skdivers use an instrument called an altimeter to track their altitude as the fall. 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 function in the diagram below. What is the altitude above sea level when the air pressure is 57,000 pascals? 59. CHEMISTRY The ph value for a substance measures how acidic or alkaline the substance is. It is given b the formula ph 52log [H ] where H is the hdrogen ion concentration (in moles per liter). Lemon juice has a hdrogen ion concentration of moles per liter. What is its ph value? 5 WORKED-OUT SOLUTIONS 504 Chapter 7 Eponential p. WS and Logarithmic Functions 5 TAKS PRACTICE AND REASONING

7 60. MULTI-STEP PROBLEM Biologists have found that an alligator s length l (in inches) and weight w (in pounds) are related b the function l ln w Graph the function. Use our graph to estimate the weight of an alligator that is 0 feet long. 6. SHORT TAKS REASONING RESPONSE The energ magnitude M of an earthquake can be modeled b Peru M (ln E) where E is the amount of energ released (in ergs). a. In 200, a powerful earthquake in Peru, caused b the slippage of two tectonic plates along a fault, released ergs. What was the energ magnitude of the earthquake? b. Find the inverse of the given function. Describe what it represents. Nazca tectonic plate Fault line South American tectonic plate 62. EXTENDED TAKS REASONING RESPONSE A stud in Florida found that the number of fish species s in a pool or lake can be modeled b the function s (log A) 3.8(log A) 2 where A is the area (in square meters) of the pool or lake. a. Graph Use a graphing calculator to graph the function on the domain 200 A 35,000. b. Estimate Use our graph to estimate the number of fish species in a lake with an area of 30,000 square meters. c. Estimate Use our graph to estimate the area of a lake that contains 6 species of fish. d. Reasoning Describe what happens to the number of fish species as the area of a pool or lake increases. Eplain wh our answer makes sense. 63. CHALLENGE The function s (log d) gives the slope s of a beach in terms of the average diameter d (in millimeters) of sand particles on the beach. Find the inverse of this function. Then use the inverse to estimate the average diameter of the sand particles on a beach with a slope of 0.2. MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson 2.2; TAKS Workbook 64. TAKS PRACTICE Which statement best describes the graph of a person s distance traveled over time? TAKS Obj. A B The person first runs, then walks. The person travels at a constant speed. Distance C The person first walks, then runs. Time D The person s speed decreases over time. REVIEW TAKS Preparation p. 408; TAKS Workbook 65. TAKS PRACTICE A window is a regular heagon. Its perimeter is 60 inches. What is the approimate area of the window? TAKS Obj. 8 F 55.9 in. 2 G in. 2 H in. 2 J 59.6 in. 2 EXTRA PRACTICE for Lesson 7.4, p. 06 Evaluate Logarithms ONLINE and Graph QUIZLogarithmic at classzone.com Functions 505

8 MIXED REVIEW FOR TEKS TAKS PRACTICE Lessons MULTIPLE CHOICE. COMPOUND INTEREST You deposit $2000 in an account that pas 4% annual interest compounded continuousl. After how man full ears will the balance first eceed $2250? TEKS 2A..F A ear B 2ears C 3ears D 6ears 2. GEOMETRIC PATTERNS When a piece of paper is folded in half, the paper is divided into two regions, each of which has half the area of the paper. If this process is repeated, the number of regions increases while the area of each region decreases. The table below shows the number of regions and the fractional area of each region after each successive fold. Which function can be used to find the fractional area A(n) of each region after n folds? TEKS 2A..D 4. PETROLEUM The amount (in billions of barrels) of oil collected b a petroleum compan drilling on the U.S. continental shelf can be modeled b ln where is the number of wells drilled. About how man barrels of oil would ou epect to be collected if 000 wells are drilled? TEKS 2A..D F. billion G 30.5 billion H 39.3 billion J 84.7 billion 5. TRANSLATIONS The graph shown below is a translation of the graph of 5 log 3. What is the equation of the graph? TEKS 2A..B (5, 2) (3, ) classzone.com (, 3) Fold number Number of regions Fractional area of each region F A(n) 5 } 2 n G A(n) 5 } (n ) n H A(n) 5 } n J A(n) 5 2 n } 2 }4 }8 }6 3. CERTIFICATES OF DEPOSIT A local bank offers certificate of deposit (CD) accounts that ou can use to save mone and earn interest. You deposit $500 into a three ear CD that pas 2% annual interest. The interest for the CD is compounded monthl. How much interest will the CD earn b the end of its term? TEKS 2A..D A $87.42 B $90.83 C $92.68 D $24.50 A 5 log 3 ( 2 2) 2 B 5 log 3 ( 2 2) C 5 log 3 ( 2 ) 2 D 5 log 3 ( 2) 2 GRIDDED ANSWER RADIOACTIVE DECAY Tritium is a radioactive substance used to illuminate eit signs. The amount of tritium disappears over time, a process called radioactive deca. If ou start with a 0 milligram sample of tritium, the number of milligrams left after t ears is given b 5 0e t. How man milligrams of tritium are left after 0 ears? Round our answer to the nearest hundredth of a milligram. TEKS 2A..D 506 Chapter 7 Eponential and Logarithmic Functions