Exponential Functions. Exponential Functions and Their Graphs. Example 2. Example 1. Example 3. Graphs of Exponential Functions 9/17/2014

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1 Eponential Functions Eponential Functions and Their Graphs Precalculus.1 Eample 1 Use a calculator to evaluate each function at the indicated value of. a) f ( ) 8 = Eample In the same coordinate place, sketch the graph of each function. a) f ( ) f ( ) 5 f ( ) 8 =1/ c) f ( ) 0. 8 =-.5 Eample In the same coordinate place, sketch the graph of each function. a) f ( ) f ( ) 5 Graphs of Eponential Functions The basic characteristics of eponential functions y = a and y = a are summarized in Figures. and.4. Graph of y = a, a > 1 Domain: (, ) Range: (0, ) y-intercept: (0, 1) Increasing -ais is a horizontal asymptote (a 0, as ). Continuous Figure. 1

2 Graphs of Eponential Functions Graph of y = a, a > 1 Domain: (, ) Range: (0, ) y-intercept: (0, 1) Decreasing Figure.4 -ais is a horizontal asymptote (a 0, as ). Continuous From Figures. and.4, you can see that the graph of an eponential function is always increasing or always decreasing. Graphs of Eponential Functions As a result, the graphs pass the Horizontal Line Test, and therefore the functions are one-toone functions. You can use the following One-to-One Property to solve simple eponential equations. One-to-One Property For a > 0 and a 1, a = a y if and only if = y. Eample 4 Solve. a) Eample 5 Describe the graph as a transformation of the graph of f ( ) 4 a) f ( ) 4 c) f ) 1 4 ( f ( ) 4 The Natural Base e In many applications, the most convenient choice for a base is the irrational number e This number is called the natural base. The function given by f() = e is called the natural eponential function. Its graph is shown in Figure.9. Eample 6 Use a calculator to evaluate the function given by f ( ) e at each value of to three decimal places. a) = 6. = -0.4 c) = -7.1 d) = 0.7 Figure.9

3 Eample 7 Applications Sketch the graph of the function s( t) 5e 0.17t Eample 8 On the day of a child s birth, a deposit of $5,000 is made in a trust fund that pays 8.5% interest. Determine the balance in this account on the child s 6 th birthday if the interest is compounded a)quarterly monthly c) continuously Eample 9 6 Ra Let Q represent the mass of radium whose half-life is 160 years. The quantity of radium present after t years is given by 1 t Q 16 / 160 a) Sketch the graph of the function over the interval from t=0 to t=5000. Determine the initial quantity (when t=0) c) Determine the quantity present after 1000 years. Logarithmic Functions Logarithmic Functions and Their Graphs Every function of the form f () = a passes the Horizontal Line Test and therefore must have an inverse function. This inverse function is called the logarithmic function with base a. Precalculus.

4 Eample 1 Use the definition of logarithmic function to evaluate each logarithm at the indicated value of. a) c) d) f ( ) log4, 16 f ( ) log, 64 f ( ) log5, 1 f ( ) log, 1 81 Logarithmic Functions The logarithmic function with base 10 is called the common logarithmic function. It is denoted by log 10 or simply by log. On most calculators, this function is denoted by. Eample Logarithmic Functions Use a calculator to evaluate the function at each value of to three decimal places. f ( ) log a) = 100 = 1/5 c) =.5 d) -4 Eample Simplify. a) log 5 1 log c) log Eample 4 Solve. a) log5 y log516 log( 4 ) log( ) c) log( 4) log 9 4

5 Eample 5 In the same coordinate plane, sketch the graph of each function. a) f ( ) 4 f ( ) log4 Graphs of Logarithmic Functions The basic characteristics of logarithmic graphs are summarized in Figure.16. Graph of y = log a, a 1 Domain: (0, ) Range: (, ) -intercept: (1, 0) Increasing One-to-one, therefore has an inverse function Figure.16 Eample 6 Sketch the graph of f ( ) log 4 Identify the vertical asymptote. Eample 7 Describe the graph as a transformation of the graph of ( ) log f a) f ( ) log 1 f ( ) log( ) The Natural Logarithmic Function Eample 8 Use a calculator to evaluate the function to three decimal places. f ( ) ln 1 a) = 7.5 = 0.4 c) = - d) = 5

6 The Natural Logarithmic Function Eample 9 Use the properties of natural logarithms to simplify. 1/ a) ln e ln8 e c) 15ln1 d) ln e 6 Eample 10 Find the domain of each function. a) f ( ) ln( ) f ( ) ln( ) c) f ( ) ln Eample 11 Students in a mathematics class were given an eam and then retested monthly with an equivalent eam. The average scores for the class are given by the human memory model f ( t) log( t 1),0 t 1 where t is time in months. a) What was the average score on the original eam? (t=0) Eample 11 f ( t) log( t 1),0 t 1 What was the average score after months? c) What was the average score after 11 months? Properties of Logarithms Precalculus. 6

7 Change of Base Change of Base Most calculators have only two types of log keys, one for common logarithms (base 10) and one for natural logarithms (base e). Although common logarithms and natural logarithms are the most frequently used, you may occasionally need to evaluate logarithms with other bases. To do this, you can use the following change-of-base formula. Eample 1 Evaluate each using the change-of-base formula with common logs. Approimate to three decimal places. a) log 16 log 5 Eample Evaluate each using the change-of-base formula with natural logs. Approimate to three decimal places. a) log 16 log 5 Properties of Logarithms Eample Write each logarithm in terms of ln and ln5. a) ln10 5 ln 7

8 Eample 4 Find the eact value of each epression without using a calculator. a) log 7 7 ln e ln e Eample 5 Epand each logarithmic epression. a) log y ln Eample 6 Condense each logarithmic epression. a) 1 log 5log( ) c) 4ln( 4) ln 1 log log( 5 ) Eponential & Logarithmic Equations Precalculus.4 Eample 1 Solve. a) 51 ln 5 ln c) d) 5 e 1 e) ln 8 f) log 8

9 Eample Solve each equation and approimate the result to three decimal places if necessary. 5 6 a) e e 4 64 Solve. 5 e Eample Eample 4 Solve the equation and approimate the result to three decimal places. 6 t Eample 5 Solve. e 7e 1 0 Eample 6 Solve. a) ln log4( ) log 4(6 ) Eample 7 Solve the equation and approimate the result to three decimal places. 6 ln 4 c) log(5 1) log 6 log 9

10 Solve. log4 6 9 Eample 8 Eample 9 Solve. log10 log10( 9) 1 Eample 10 You have deposited $1000 in an account that pays 6.5% compounded continuously. How long will it take your money to double? Eample 11 The number y of endangered animal species on a protected wildlife preserve from 1990 to 004 can be modeled by y lnt 10 t 4, where t represents the year, with t=10 corresponding to During which year did the number of endangered animal species reach 4? Introduction Eponential & Logarithmic Models Precalculus.5 The five most common types of mathematical models involving eponential functions and logarithmic functions are as follows. 1. Eponential growth model: y = ae b, b 0. Eponential decay model: y = ae b, b 0. Gaussian model: 4. Logistic growth model: y = y = ae ( /c 5. Logarithmic models: y = a + b ln, y = a + b log 10

11 Introduction Introduction The basic shapes of the graphs of these functions are shown in Figure.. Logistic growth model Natural logarithmic model Common logarithmic model Eponential growth model Eponential decay model Gaussian model Figure. Figure. Eample 1 The population P of a city is given by 0.055t P 95,00e where t=0 represents 001. According to this model, when did the population reach 150,000? Eample In a research eperiment, a population of fruit flies is increasing according to the law of eponential growth. After days there are 15 fruit flies, and after 4 days there are 50 flies. How many flies will there be after 6 days? Eample Estimate the age of a newly discovered fossil in which the ratio of carbon 14 to carbon 1 is 1 R Gaussian Models The Gaussian models are of the form y = ae ( /c. This type of model is commonly used in probability and statistics to represent populations that are normally distributed. The graph of a Gaussian model is called a bell-shaped curve. 11

12 Eample 4 The average value of a population can be found from the bell-shaped curve by observing where the maimum y value of the function occurs. The value corresponding to the maimum y value of the function represents the average value of the independent variable in this case,. Last year, the math scores for students in a particular math class roughly followed the normal distribution given by ( 74) 114 y 0.099e,0 110 where is the math score. Sketch the graph of this function, and use it to estimate the average math score. Logistic Growth Models Some populations initially have rapid growth, followed by a declining rate of growth, as indicated by the graph in Figure.40. One model for describing this type of growth pattern is the logistic curve given by the function Logistic Growth Models An eample is a bacteria culture that is initially allowed to grow under ideal conditions, and then under less favorable conditions that inhibit growth. where y is the population size and is the time. Figure.40 A logistic growth curve is also called a sigmoidal curve. Eample 5 Eample 5 On a college campus of 7500 students, one student returns from vacation with a contagious and long-lasting virus. The spread of the virus is modeled by 7500 y, t 0 0.9t e where y is the total number of students affected after t days. The college will cancel classes when 0% or more of the students are infected y, t 0 0.9t e a) How many students will be infected after 4 days? After how many days will the college cancel classes? 1

13 Eample 6 On the Richter scale, the magnitude R of an I earthquake of intensity I is given by R log I where I 1 0 is the minimum intensity used for comparison. Find the magnitude R of an earthquake of intensity I. a) I 68,400,000 I 4,75,