7.4. Characteristics of Logarithmic Functions with Base 10 and Base e. INVESTIGATE the Math

Size: px

Start display at page:

Download "7.4. Characteristics of Logarithmic Functions with Base 10 and Base e. INVESTIGATE the Math"

Avice Nichols
6 years ago
Views:

1 7. Characteristics of Logarithmic Functions with Base 1 and Base e YOU WILL NEED graphing technolog EXPLORE Use benchmarks to estimate the solution to this equation: logarithmic function A function of the form 5 a log b where b., b 1, and a, and a and b are real numbers. GOAL Investigate the characteristics of logarithmic functions with base 1 and base e from graphs and equations. INVESTIGATE the Math Rand came across a new tpe of function in his chemistr class. He learned that the ph of a solution is determined b a logarithmic function. He wondered how the characteristics of these functions differ from the characteristics of polnomial and eponential functions. Rand noticed that he had a log button on his calculator, so he looked in his user manual and discovered that this was a base 1 logarithm. Using this button, he created a table of values and a graph for each of the functions shown below to investigate the similarities and differences. f1 5 log 1 g1 5 log 1 h1 5 5 log 1 1 undefined 1 undefined 1 undefined undefined undefined undefined f() log 1 g() log h() 5 log 1 7 Chapter 7 Eponential and Logarithmic Functions NEL

2 ? What are the characteristics of logarithmic functions of the form 5 a log b, where b 5 1 or b 5 e, and a is a real number? A. Use a graphing calculator to graph the logarithmic function 5 log 1. On the same aes, graph 5 1. How are these two graphs related? B. Rand choose three functions of the form 5 a log 1, where a.. On a new screen, graph Rand s functions, shown on the previous page. C. Eamine the graph of each function, and state the following characteristics: the number of -intercepts the -intercept the end behaviour the domain the range D. Eamine the graphs ou created in part B. As ou move from left to right along the -ais, do the -values of each function increase or decrease? Then eamine the table of values for each function on our graphing calculator. What happens to the -values as the -values increase? Tip Communication The epression log 1 is known as the common logarithm or a logarithm with a base of 1. The epression is often written without the 1, so the two functions 5 log 1 and 5 log are equivalent. Tip Communication The function 5 log 1 is equivalent to 5 1, so a logarithm is an eponent. The meaning of log 1 is the eponent that must be applied to base 1 to get the value of. For eample, log E. Choose three new functions of the form 5 a log 1, where a,. Repeat parts C and D for these new functions. F. On a new screen, graph the natural logarithmic function 5 ln. On the same aes, graph 5 e. How are these two graphs related? G. On a new screen, graph the function 5 ln and two other functions of the form 5 a ln, where a.. Eamine the graph of each function, and state the following characteristics: the number of -intercepts the -intercept the end behaviour the domain the range Tip Communication A logarithm with base e is called the natural logarithm and is written as ln. The functions 5 log e, 5 ln, and 5 e are equivalent. H. Eamine the graphs ou created in part G. As ou move from left to right along the -ais, do the -values of each function increase or decrease? Then eamine the table of values for each function on our graphing calculator. What happens to the -values as the -values increase? I. Choose three new functions of the form 5 a ln, where a,. Repeat parts G and H for these new functions. NEL 7. Characteristics of Logarithmic Functions with Base 1 and Base e 75

3 Reflecting J. Based on our observations, which characteristics of the graphs of logarithmic functions are similar to the characteristics of graphs of eponential functions ou have studied? K. Based on our observations, which characteristics of the graphs of logarithmic functions differ from the characteristics of graphs of eponential functions ou have studied? L. Can ou predict the end behaviour of functions of the form 5 a log or 5 a ln based on the parameter a? Eplain. M. In logarithmic functions of the form 5 a log b, did changing the base from b 5 1 to b 5 e cause an of the characteristics to change? Eplain. APPLY the Math eample 1 Connecting the characteristics of an increasing logarithmic function to its equation and graph Predict the -intercept, the number of -intercepts, the end behaviour, the domain, and the range of the following function: 5 15 log Use the equation of the function to make our predictions. Verif our predictions using graphing technolog. Sid s Solution The function 5 15 log is a logarithmic function of the form 5 a log b, with a 5 15 and b 5 1. Common Characteristics -intercept: 1 Number of -intercepts: Domain: 5., [ R Range: 5 [ R I knew that this is a common logarithmic function since the base is not written. Its base is 1. I determined the values of the parameters a and b. I predicted the characteristics of the function from its equation. I started with the common characteristics I knew for all logarithmic functions of the form 5 a log b, where b Chapter 7 Eponential and Logarithmic Functions NEL

4 Unique Characteristics End behaviour: The curve etends from quadrant IV to quadrant 1. Since the parameter a is greater than, I knew that the function increases as ou move from left to right along the -ais. I used a graphing calculator to create a table of values and a graph to verif m predictions. M predictions were correct. Your Turn Predict the -intercept, the number of -intercepts, the end behaviour, the domain, and the range of this function: 5 5 log Use the equation of the function to make our predictions. Verif our predictions using graphing technolog. eample Connecting the characteristics of a decreasing natural logarithmic function to its equation and graph Predict the -intercept, the number of -intercepts, the end behaviour, the domain, and the range of the following function: 5 ln Use the equation of the function to make our predictions. Verif our predictions using graphing technolog. NEL 7. Characteristics of Logarithmic Functions with Base 1 and Base e 77

5 Anne s Solution The function 5 ln is a logarithmic function of the form 5 a log b with a base of e. In this function, a 5 and b 5 e. Common Characteristics -intercept: 1 Number of -intercepts: Domain: 5., [ R Range: 5 [ R Unique Characteristics End behaviour: The curve etends from quadrant I to quadrant IV. I knew that 5 ln is the natural logarithmic function, which is equivalent to 5 log e, so I knew the base of the logarithmic function is.71. I determined the values of the parameters a and b. I predicted the characteristics of the function from its equation. I started with the common characteristics I knew for all logarithmic functions of the form 5 a log b, where b 5 e. Since the paameter a is less than, I knew that the function decreases as ou move from left to right along the -ais. I used a graphing calculator to create a table of values and a graph. M predictions were correct. Your Turn Predict the -intercept, the number of -intercepts, the end behaviour, the domain, and the range of this function: 5 1 ln Use the equation of the function to make our predictions. Verif our predictions using graphing technolog. 7 Chapter 7 Eponential and Logarithmic Functions NEL

6 eample 3 Matching equations of eponential and logarithmic functions with their graphs Which function matches each graph below? Provide our reasoning. i) 5 51 ii) iii) 5 log iv) 5 ln a) c) b) d) Errn s Solution Eponential functions: I know that graphs b) and d) are eponential functions of the form 5 a1b, where a., b., and b 1, since both etend from quadrant II to quadrant I. I also know that functions i) and ii) are eponential, since the eponent in each equation is a variable. 5() Since b. in the function 5 51, I knew that this function matches the graph of the increasing function. NEL 7. Characteristics of Logarithmic Functions with Base 1 and Base e 79

7 (.1) Since b, in the function 5 1.1, I knew that this function matches the graph of the decreasing function intercept: intercept: Function i) matches graph b) and function ii) matches graph d). Logarithmic functions: I know that graphs a) and c) are logarithmic, since the etend from quadrant I to quadrant IV and from quadrant IV to quadrant I, respectivel. I also know that functions iii) and iv) are logarithmic. log I knew that I was right since the -intercepts for the equations match the -intercepts on the graphs. Since a. in the function 5 log, I knew that this function matches the graph of the increasing function. ln Since a, in the function 5 ln, I knew that this function matches the graph of the decreasing function. Function iii) matches graph c) and function iv) matches graph a). Your Turn Jordan claims that she can tell which graphs are eponential and which graphs are logarithmic b eamining the domain of each function. Do ou agree or disagree? Eplain. Chapter 7 Eponential and Logarithmic Functions NEL

8 In Summar Ke Ideas A logarithmic function has the form f1 5 a log b, where b., b 1, and a, and a and b are real numbers. All logarithmic functions of the form f1 5 a log and f1 5 a ln have these characteristics: -Intercept 1 Number of -Intercepts End Behaviour Domain Range The curve etends from quadrant IV to quadrant I or quadrant I to quadrant IV. 5., [ R 5 [ R All logarithmic functions of the form f1 5 a log and f1 5 a ln have these unique characteristics: - If a., the function increases. - If a,, the function decreases. Need to Know The graph of a logarithmic function of the form f1 5 a log or f1 5 a ln will look like one of the following cases: Case 1: an increasing function, where a. Case : a decreasing function, where a, The graph of 5 log is a reflection of the graph of 5 1 about the line 5. - log - 1 The graph of 5 ln is a reflection of the graph of 5 e about the line 5. - ln - e NEL 7. Characteristics of Logarithmic Functions with Base 1 and Base e 1

9 1 log check Your Understanding 1. State the -intercept, the number of -intercepts, the end behaviour, the domain, and the range of the function shown on the left.. Identif which of these graphs are logarithmic functions. Eplain how ou know. a) d) b) e) c) f ) Chapter 7 Eponential and Logarithmic Functions NEL

10 3. For the functions in question that are logarithmic, state the -intercept, the number of -intercepts, the end behaviour, the domain, and the range. Also state whether the parameter a in the equation of the function is greater than zero or less than zero, and eplain how ou know. PRACTISING. Use graphing technolog to graph the functions 5 log and 5 log. Compare the graphs of the functions. Consider the -intercept, the number of -intercepts, the end behaviour, the domain, the range, and whether the are increasing or decreasing. 5. Predict the -intercept, the number of -intercepts, the end behaviour, the domain, the range, and whether each logarithmic function is increasing or decreasing, based on its equation. Verif our predictions using graphing technolog. a) f log b) f log c) f ln d) f ln e) f 1 5. log f) f ln. State three reasons wh ou know that the following graph represents a logarithmic function of the form 5 a log NEL 7. Characteristics of Logarithmic Functions with Base 1 and Base e 3

11 7. Sketch the graph of the logarithmic function with each set of characteristics. a) -intercept: 1 Number of -intercepts: End behaviour: The curve etends from quadrant IV to quadrant I. Domain: 5., [ R Range: 5 [ R b) -intercept: 1 Number of -intercepts: End behaviour: The curve etends from quadrant I to quadrant IV. Domain: 5., [ R Range: 5 [ R. Match each function with its corresponding graph. Provide our reasoning. i) 5. log ii) 5 3 log iii) 5 iv) a) - - c) - - b) d) Chapter 7 Eponential and Logarithmic Functions NEL

12 9. Holl claims that she can distinguish between the graph of an eponential function of the form 5 a1b, where a., b., and b 1, and the graph of a logarithmic function of the form 5 a log, where a, based entirel on the number of -intercepts shown. Do ou agree or disagree? Eplain. 1. The ph of a solution, p(), can be determined using the function p1 5 log where represents the concentration of hdrogen ions in the solution, in moles per litre. Use the equation of the function to predict what happens to the ph of a solution as the concentration of hdrogen ions increases. Verif our prediction using graphing technolog. 11. Sound intensit is measured on the decibel scale. A decibel is 1 1 of a bel, named after Aleander Graham Bell. The intensit of a sound (number of decibels) can be determined using the function D1 5 1 log 11 1 # where represents the energ of the sound in watts per square centimetre (watts/cm ). Use the equation of the function to predict what happens to the number of decibels, D(), as the energ of the sound,, increases. Verif our prediction using graphing technolog. A recording of speech. Like other sounds, speech can be measured b pitch (top graph) and b intensit (bottom graph). NEL 7. Characteristics of Logarithmic Functions with Base 1 and Base e 5

13 Time (ears) t Radioactive Deca of Cesium-137 P Percent of original cesium Cesium-137 is created in nuclear reactors. It decas naturall over time. The approimate time for cesium-137 to deca is given b the function t 5 1. log P where t represents the time in ears and P represents the percent of the original amount of cesium-137, as a decimal, that remains. The function is graphed at the left. a) State the following characteristics of the graph: the intercepts the domain and range within the contet of this problem whether the function is increasing or decreasing the value of the function at P 5 1 b) A sample of cesium-137 is taken from a reactor. Estimate the time when 5% of the sample will remain. c) Estimate the time when 5% of the sample will remain. Closing 13. Consider the following functions: 5 a log and 5 a ln a) For what values of a is each function decreasing? b) Do these functions have an unrestricted domain? Eplain. c) Do these functions have an unrestricted range? Eplain. Etending 1. The largest known earthquake in Saskatchewan occurred in 199. Based on recorded observations, the earthquake is believed to have measured approimatel 5.5 on the Richter scale. The magnitude, M, of an earthquake that is T times more intense than an earthquake registering 5.5 on the Richter scale is given b the following function: M 5 log T For ever 1 unit increase on the Richter scale, the intensit of an earthquake increases b a factor of 1. a) Use technolog to graph the function, and state the following characteristics of the graph: the intercepts the domain and range within the contet of this problem whether the function is increasing or decreasing the value of the function at T 5 1 b) Determine the magnitude of an earthquake that is 5 times more intense than the Saskatchewan earthquake. c) In 11, an earthquake that measured 9. on the Richter scale occurred off the coast of Japan, creating a devastating tsunami. About how man times more intense was this earthquake than the Saskatchewan earthquake? Use our graph to estimate. Chapter 7 Eponential and Logarithmic Functions NEL

14 15. Christie has $3 invested in a savings account that earns %/a. The approimate time for her investment to reach a given future value is modelled b the function t log A.1 where t is the time in ears and A is the future value. The function is graphed below. Value of Savings t Time, t (ears) Future value, A ($) A a) State the following characteristics of this function: the intercepts the domain and range within the contet of this problem whether the function is increasing or decreasing b) How long, in ears, will it take for Christie s investment to reach $1? c) How long, in ears, will it take for Christie s investment to double? NEL 7. Characteristics of Logarithmic Functions with Base 1 and Base e 7

2. Tell whether the equation or graph represents an exponential growth or exponential decay function.

2. Tell whether the equation or graph represents an exponential growth or exponential decay function. Name: Date: Period: ID: 1 Unit 9 Review Eponents & Logarithms NO GRAPHING CALCULATOR 1. Under each function, write es if it is an eponential function. If the answer is no, write an eplanation wh not. a)