Presentation on theme: "Exponential and Logarithmic Functions"— Presentation transcript:
1Exponential and Logarithmic Functions Chapter 13Exponential and Logarithmic Functions
2Section 13.1: Exponential Functions and Their Graphs MAT 205 SP 2009Section 13.1: Exponential Functions and Their GraphsDefinition of an Exponential FunctionThe exponential function with base b is denoted bySo, in an exponential function, the variable is in the exponent.
3Exponential Functions Which of the following are exponential functions?
4Graphs of Exponential Functions They can be broken into two categories—exponential growth, andexponential decay (decline).
5The Graph of an Exponential Growth Function We will look at the graph of an exponential function that increases as x increases, known as the exponential growth function. It has the formExample: y = 2xxy-5-4-3-2-1123y = 2xNotice the rapid increase in the graph as x increasesThe graph increases slowly for x < 0.y-intercept is (0, 1)Horizontal asymptote is y = 0.
7The Graph of an Exponential Decay (Decline) Function We will look at the graph of an exponential function that decreases as x increases, known as the exponential decay function. It has the formExample: y = 2-xy = 2-xxy-3-2-112345Notice the rapid decline in the graph for x < 0.The graph decreases more slowly as x increases.y-intercept is (0, 1)Horizontal asymptote is y = 0.
8Graphs of Exponential Functions Notice that f(x) = 2x and g(x) = 2-x are reflections of one another about the y-axis.Both graphs have y-intercept ___________ and horizontal asymptote ________ .The domain of f(x) and g(x) is _________; the range is _______.
9Graphs of Exponential Functions MAT 205 SP 2009Graphs of Exponential FunctionsAlso, note that , applying the properties of exponents.So an exponential function is a decay function ifThe base b is greater than one and the function is written as f(x) = b-x-OR-The base b is between 0 and 1 and the function is written as f(x) = bx
10Graphs of Exponential Functions MAT 205 SP 2009Graphs of Exponential FunctionsExamples:In this case, b = 0.25 (0 < b < 1).In this case, b = 5.6 (b > 1).
11Natural base eIt may seem hard to believe, but when working with exponents and logarithms, it is often convenient to use the irrational number e as a base.The number e is defined asThis value approaches as x approaches infinity.
12Evaluating the Natural Exponential Function To evaluate the function f(x) = ex, we will use our calculators to find an approximation. You should see the ex button on your graphing calculator (Use ).Example:Given , find f(3) and f(-0.5) to 3 decimal places.≈ ____________≈ _______________
13Graphing the Natural Exponential Function Growth or decay?Domain:Range:Asymptote:x-intercept:y-intercept:List four points that are on the graph of f(x) = ex.
14Graphing the Natural Exponential Function Determine the following:Growth or decay?Domain:Range:Asymptote:x-intercept:y-intercept:
15ExampleThe population of a town is modeled by the function where t = 0 corresponds to 1990 and P is the town’s population in thousands.According to the model, what was the town’s population in 1990?According to the model, what was the town’s population in 2008?
16MAT 205 SP 2009Example (continued)Graph the function on your calculator and determine in which year the town’s population reached 75,000 people.How would we solve this algebraically??
17Section 13.2: Logarithmic Functions MAT 205 SP 2009Section 13.2: Logarithmic FunctionsNow that you have studied the exponential function, it is time to take a look at its INVERSE: the LOGARITHMIC FUNCTION.In the exponential function, the independent variable was the exponent. So we substituted values into the exponent and evaluated it for a given base. For example, for f(x) = 2xf(3) =
18Logarithmic Functions MAT 205 SP 2009Logarithmic FunctionsFor the inverse (logarithmic) function, the base is given and the answer is given, so to evaluate a logarithmic function is to find the exponent.That is why I think of the logarithmic function as the “Guess That Exponent” function.Warm Up: Give the value of ? in each of the following equations.
19Logarithmic Functions (continued) For example, to evaluate log28 means to find the exponent such that 2 raised to that power gives you 8.
20Logarithmic Functions (continued) The following definition demonstrates this connection between the exponential and the logarithmic function.Definition of an Logarithmic FunctionFor y > 0, b > 0, and b ≠ 1,If y = bx , then x = logbyy = bx is the exponential formx = logby is the logarithmic formWe read logby as “log base b of y”.
21MAT 205 SP 2009Subliminal Message:The exponential and logarithmic functions of the same base are inverses.
22Converting Between Exponential and Logarithmic Forms If y = bx, then x = logbyI. Write the logarithmic equation in exponential form.a)b)II. Write the exponential equation in logarithmic form.
23The plan is to convert to exponential form. Evaluating Logarithms w/o a CalculatorTo evaluate logarithmic expressions by hand, we can use the related exponential expression.Example:Evaluate the following logarithms:The plan is to convert to exponential form.
28Graphs of Logarithmic Functions Example: Graph f(x) = 2x and g(x) = log2x in the same coordinate plane.Solution: To do this, make a table of values for f(x) and then switch the x and y coordinates to make a table of values for g(x).
30Graphs of Logarithmic Functions (continued) Notice how the domain and range of the inverse functions are switched.The exponential function hasDomain: ____________Range: ____________Horizontal asymptote: _________The logarithmic function hasDomain: __________Range: ___________Vertical asymptote: __________f(x) = 2xg(x)= log2xy =x
31Back in my day, we used log tables and slide rules!” Section 13.4: Evaluating Common Logarithms with a CalculatorNot all logarithmic expressions can be evaluated easily by hand. In fact, most cannot.For example, to evaluate is to find x such that 2x = 175.This is not a simple task. In fact, the answer is irrational. For these types of problems, we will use the calculator.“Calculators??Back in my day, we used log tables and slide rules!”
32Evaluating Common Logarithms with a Calculator (continued) The calculator, however, only calculates two different base logarithms—the common logarithm and the natural logarithm.I. The COMMON LOGARITHM is the logarithmic function with base 10.On the TI-83/84, look for the button. This is used to evaluate the common log (base 10) only.Example:Evaluate f(x)=log10x for x = 400. Round to four decimal places.Solution:f(400) = log Answer: ___________LOGLOGENTER
33Antilog of the Common Log We can also find a number given its logarithm.We say that N is the antilog ofWe use 2ND LOG [10x]Example: log N =N = _____________________________
34Application of the Logarithm ExampleMeasured on the Richter , the magnitude of an earthquake of intensity I is defined to be R = Log(I/I0), where I0 is a minimum level for comparison. What is the Richter scale reading for the 1995 Philippine earthquake for which I=20,000,000 I0?
35Section 13.5: The Natural Logarithmic Function In section 13.1, we saw the natural exponential function with base e. Its inverse is the natural logarithmic function with base e.Instead of writing the natural log as logex, we use the notation ln x, which is read as “the natural log of x” and is understood to have base e.
36The Natural Logarithmic Function To evaluate the natural log using the TI-83/84, use the button.ExampleEvaluate the function f(x) = ln x ata) x = 1.5b) x = -2.3LNThis means that ______________________________________
37Graph of the Natural Exponential and Natural Logarithmic Function f(x) = ex and g(x) = ln x are inverse functions and, as such, their graphs are reflections of one another in the line y = x.
38Antilog of the Natural Logarithm We say that N is the antilog ofWe use 2ND LOG [ex]Example: ln N =N = _____________________________2NDLN
39Change-of-Base Formula I mentioned that the calculator only has two types of log keys, the COMMON LOG (BASE 10) and the NATURAL LOG (BASE e). It’s true that these two types of logarithms are used most often, but sometimes we need to evaluate logarithms with bases other than 10 or e.To do this on the calculator, we use a CHANGE-OF-BASE FORMULA. We will convert the logarithm with base a into an equivalent expression involving common logarithms or natural logarithms.
40Change-of-Base Formula (continued) Let a, b, and x be positive real numbers such that a 1 and b 1. Then logbx can be converted to a different base using any of the following formulas.
41Change-of-Base Formula Examples* Use the change-of-base formula to evaluate log7264using common logarithmsusing natural logarithms.Solution:The result is the same whether you use the common log or the natural log.
42Change-of-Base Formula Examples Use the change-of-base formula to evaluatea) b)
43Graph of the Logarithmic Function with base b MAT 205 SP 2009Graph of the Logarithmic Function with base ba) Graph on the calculator.b) Graph its inverse on the calculator.
44Section 13.3: Properties of Logarithms MAT 205 SP 2009Section 13.3: Properties of LogarithmsLet b be a positive real number such that b 1, and let n, x, and y be real numbers.Base b Logarithms Natural Logarithms
46Use the properties of logs to EXPAND each of the following expressions into a sum, difference, or multiple of logarithms:
47Again! Use the properties of logs to EXPAND each of the following expressions into a sum, difference, or multiple of logarithms:
48This is fun! Use the properties of logs to EXPAND each of the following expressions into a sum, difference, or multiple of logarithms:
49Try these! Use the properties of logs to CONDENSE each of the expressions into a logarithm of a single quantity:
50Properties of Logarithms Rock! Use the properties of logs to CONDENSE each of the expressions into a logarithm of a single quantity:
51One more! Use the properties of logs to CONDENSE each of the expressions into a logarithm of a single quantity:
52Solving EXPONENTIAL Equations: Part I MAT 205 SP 2009Section 13.6: Solving Exponential and Logarithmic EquationsSolving EXPONENTIAL Equations: Part II. Using the One-to-One Property If you can write the equation so that both sides are expressed as powers of the SAME BASE, you can use the property bx = by if and only if x = y. Example: Solve 4x-2 = 64
53Solving EXPONENTIAL Equations: Part II MAT 205 SP 2009Solving EXPONENTIAL Equations: Part IIII. By Taking the Logarithm of Each SideISOLATE the exponential term on one side of the equation.TAKE THE COMMON OR NATURAL LOG of each side of the equation.USE THE PROPERTIES OF LOGARITHMS to remove the variable from the exponent.SOLVE for the variable. Use the calculator to evaluate the resulting log expression.
54Solving EXPONENTIAL Equations Example: Solve 3(54x+1) -7 = Give answer to 3 decimal places.
55Solving LOGARITHMIC Equations: Part I I. Using the One-to-One Property If you can write the equation so that both sides are expressed as SINGLE logarithms with the SAME BASE, you can use the property logbx = logby if and only if x = y.
56Solving LOGARITHMIC Equations: Part I Example of one-to-one property:Solve log3x + 2log35 = log3(x + 8)
57Solving Logarithmic Equations: Part II II. By Rewriting in Exponential FormUSE THE PROPERTIES OF LOGARITHMS to combine log expressions into a SINGLE log expression, if necessary.ISOLATE the logarithmic expression on one side of the equation.Rewrite the equation in EXPONENTIAL FORM.SOLVE the resulting equation for the variable.CHECK the solution in the original equation either graphically or algebraically
59Solving Exponential and Logarithmic Equations GRAPHICALLY Remember, you can verify the solution of any one of these equations by finding the graphical solution using your TI-83/84 calculator.Enter the left hand side of the original equation as y1,Enter the right side as y2, andFind the point at which the graphs intersect. Below is the graphical solution for the last example.The x-coordinate of the intersection point is approximately , confirms our algebraic solution.
60I. Solve each of the following EXPONENTIAL equations I. Solve each of the following EXPONENTIAL equations. Round to 4 decimal places, if necessary.
71MAT 205 SP 2009ApplicationsExample: How long will it take $25,000 to grow to $500,000 if it is invested at 9% annual interest compounded monthly? Round to the nearest tenth of a year.Formula:
72Another Example:The population of Asymptopia was 6500 in 1985 and has been tripling every 12 years since then. If this rate continues, when will the population reach 75,000?Let t represent the number of years since 1985P(t) represents the population after t years.
73Drug medication: The formula can be used to find the number of milligrams D of a certain drug that is in a patient’s bloodstream h hours after the drug has been administered. When the number of milligrams reaches 2, the drug is to be administered again. What is the time between injections? Round to the nearest tenth of an hour.
74A Logarithmic Model:The loudness L, in bels (named after ?), of a sound of intensity I is defined to bewhere I0 is the minimum intensity detectable by the human ear.The bell is a large unit, so a subunit, the decibel, is generally used. For L, in decibels, the formula is
75A Log Model (cont)Find the loudness, in decibels, for each sound with the given intensity.LibraryDishwasherLoud muffler
76A Log Model (cont)If the front rows of a rock concert has a loudness of 110 dB and normal conversation has a loudness of 60 dB , how many times greater is the intensity of the sound in the front rows of a rock concert than the intensity of the sound of normal conversation?
77x y Section 13.7: Graphs on Log and Semi-log Paper MAT 205 SP 2009Section 13.7: Graphs on Log and Semi-log PaperLet’s say we want to plot the graph y = 5xxy-112345The detail for values of x less than 3 is nearly imperceptible.
78MAT 205 SP 2009Often times, we want to model data that require that small variations at one end of the scale are visible, while large variations at the other end are also visible.To graph functions where one or both of the variables have a wide change in values, we can use a logarithmic scale.This type of scale is marked off in distances that are proportional to the logarithm of the values being represented.The distances between integers on a log scale are not equal, but will give us a better way to show a greater range of values.
79The horizontal scale has equal spacing between the lines MAT 205 SP 2009If we want to show a large range of values for only one of your variables, we will use SEMI-LOG paper. Semi-log paper has two scales:The horizontal scale has equal spacing between the linesThe vertical scale does not have equal spacing between the lines. It uses a logarithmic scale.
80MAT 205 SP 2009Semi-log paper allows you to graph exponential data without having to translate your data into logarithms—the paper does it for you. The scale of semi-log paper has cycles. Below is what is known as 3-cycle semi-log graph paper.On the vertical scale, the powers of ten are evenly spaced.On the horizontal scale, the numbers along the axis are evenly spaced.
81x y Let’s see the graph of y = 5x on semi-log paper. -1 0.2 1 5 2 25 3 MAT 205 SP 2009Let’s see the graph of y = 5x on semi-log paper.xy-10.215225312546253125
82MAT 205 SP 2009Semi-log paper is often used to transform a nonlinear data relation into a linear one.If a function makes a STRAIGHT LINE when graphed on semi-log graph paper, we call it an EXPONENTIAL FUNCTION.
83On log-log paper, both axes are marked with a logarithmic scale. MAT 205 SP 2009On log-log paper, both axes are marked with a logarithmic scale.
84x y Example: Create a log-log plot of the function y = 0.5x3. 0.5 1 2 MAT 205 SP 2009Example: Create a log-log plot of the function y = 0.5x3.xy0.5125
85MAT 205 SP 2009Notice that the graph of y = 0.5x3 on log-log paper is a straight line.An equation in the form of y = axb is called a POWER FUNCTION. If you plot the data points of a power function on log-log paper, it appears LINEAR.