Chapter 3 Exponential & Logarithmic Functions

Chapter 3 Exponential & Logarithmic FunctionsSection 3.1 Exponential Functions & Their GraphsDefinition of Exponential Function:The exponential function f with base a is denotedf(x) ax where a 0, a 1 and x is any real number.Example 1: Evaluating Exponential ExpressionsUse a calculator to evaluate each expressiona. 2-3.1b. 2-πExample 2: Graphs of y axIn the same coordinate plane, sketch the graph of each function.a. f(x) 2xb. g(x) 4x1

Example 3: Graphs of y a-xIn the same coordinate plane, sketch the graph of each function.a. F(x) 2-xb. G(x) 4-xCompare the functions in examples 2 and 3.F(x) 2-x andGraph of y axG(x) 4-x Graph of y ontalAsympotote2

Example 4: Sketching Graphs of Exponential FunctionsCompare the following graphs to g(x) 3xa. g(x) 3x 1c.k(x) -3xb. h(x) 3x – 2d. j(x) 3-xThe Natural Base ee 2.71828. is called the natural base.The function f(x) ex is called the natural exponential function.Example 5: Evaluating the Natural Exponential FunctionUse a calculator to evaluate each expression.a. e-2b. e-1c. e1d. e23

Example 6: Graphing Natural Exponential FunctionsSketch the graph of each natural exponential function.a. f(x) 2e0.24xb. g(x) e-0.58xFormulas for Compound InterestAfter t years, the balance A in an account with principal P and annual interest rate r (in decimal form)is given by the following formulas:1) For n compoundings per year: A P(1 / )nt2) For continuous compounding: A PertExample 7: Compounding n Times and ContinuouslyA total of 12,000 is invested at an annual rate of 9%. Find the balance after 5 years if it iscompoundeda) Quarterlyb) Continuously4

Example 8: Radioactive DecayIn 1986, a nuclear reactor accident occurred in Chernobyl in what was then the Soviet Union. Theexplosion spread radioactive chemicals over hundreds of square miles, and the governmentevacuated the city and the surrounding area. To see why the city is now uninhabited, consider thefollowing model:P 10e-0.0000845tThis model represents the amount of plutonium that remains (from an initial amount of 10 pounds)after t years. Sketch the graph of this function over the interval from t 0 to t 100,000. How muchof the 10 pounds will remain after 100,000 years?5

Section 3-2: Logarithmic Functions and Their GraphsLOGARITHMIC FUNCTIONSRemember From Section 1.6 – a function has an inverse if it passes the Horizontal Line Test (nohorizontal line intersects the graph more than once). Exponential functions of the formf (x) a xpass the Horizontal Line Test, which means theymust have an inverse. The inverse of an exponential function is called the logarithmic function with base a.Try and start finding the inverse of: f ( x) axWhere do you run into trouble?DEFINITION OF LOGARITHMIC FUNCTION:Forx 0 and 0 a 1,y log a x if and only if x a y .The function given byf (x) log a xis called the logarithmic function with base a.6

Remember A logarithm is an exponent! This means thatbe raised to obtain x. For instance,log a xis the exponent to which a mustlog2 8 3 because 2 must be raised to the third power to get 8.Example 1: Evaluating Logarithms1 100a. log 2 32 d.log10b.log 3 27 e.log 3 1 c.log 4 2 f.log 2 2 The logarithmic function with base 10 is called the common logarithmic function.Example #2: Evaluating Logarithms on a CalculatorUse a calculator to evaluate each expression.a.log10 10b.2log10 2.5c.log10 ( 2)7

PROPERTIES OF LOGARITHMS:1.log a 1 0 because 2.log a a 1 because 3.loga a x x4. Ifbecause loga x log a y , then x y .Example #3: Using Properties of LogarithmsSolve each equation for x. List which property you used.a.)log 2 x log 2 3b.)log4 4 x8

GRAPHS OF LOGARITHMIC FUNCTIONSRemember The graphs of inverse functions are reflections of each other in the liney x.Example #4: Graphs of Exponential and Logarithmic FunctionsIn the same coordinate plane, sketch the graph of each function.a.f (x) 2 xb.g(x) log2 xExample #5: Sketching the Graph of a Logarithmic FunctionSketch the graph of the common logarithmic functionf (x) log10 x .9

Basic Characteristics of Logarithmic Graphs:Graph ofy log a x , a 1 Domain: Range: Intercept: Increasing or decreasing? Vertical asymptote: Reflection:Example #6: Sketching the Graphs of Logarithmic FunctionsSketch each of the following, referring tof (x) log10 x .Be sure to describe thetransformation.a.g(x) log10 (x 1)b.h(x) 2 log10 x10

THE NATURAL LOGARITHMIC FUNCTIONThe logarithmic function with base e is the natural logarithmic function and is denoted by the specialsymbolln x , read as “el en of x.”PROPERTIES OF NATURAL LOGARITHMS:1.ln1 0 because 2.lne 1 because 3.ln e x x4. Ifbecause ln x ln y , then x y .Example #7: Using Properties of Natural Logarithms1 a. lneb.lne 2 c.ln e 0 d.2lne Example #8: Evaluating the Natural Logarithmic FunctionUse a calculator to evaluate each expression.a.ln2b.ln0.3c.ln e 2d.ln( 1)11

Example #9: Finding the Domain of Logarithmic FunctionsFind the domain of each function.a.f (x) ln(x 2)b.g(x) ln(2 x)c.h(x) ln x 2Example #10: Human Memory ModelStudents participating in a psychological experiment attended several lectures on a subject and weregiven an exam. Every month for a year after the exam, the students were retested to see how much ofthe material they remembered. The average scores for the group are given by the human memorymodelf (t) 75 6ln(t 1) ,0 t 12where t is the time in months.a. What was the average score on the original (t 0) exam?b. What was the average score at the end of t 2 months?c. What was the average score at the end of t 6 months?12

Section 3-3 Properties of LogarithmsYou know that you can only put ln or log10 into your calculator. In order to evaluate other logs, youwill need the change-of-base formula.Change-of-Base FormulaLet a, b, and x be positive real numbers such that a 1 and b 1.Then loga x is given byloga x Example 1: Changing Bases Using Common Logarithmsa. log430 b. log214 Example 2: Changing Bases Using Natural Logarithmsa. log430 b. log214 Properties of LogarithmsYou know from the previous section that the logarithmic function is the inverse of the exponentialfunction. Thus, it makes sense that properties of exponents should have corresponding propertiesinvolving logarithms.Properties of LogarithmsLet a be a positive number such that a 1, and let n be a real number. If u and v are positive realnumbers, the following properties are true:1. logauv logau logav1. ln (uv) ln u ln v2. loga logau - logav2. ln ln u – ln v3. logaun nlogau3. ln un n ln u13

Example 3: Using Properties of LogarithmsWrite the logarithm in terms of ln 2 and ln 3a. ln 6b. lnExample 4: Using Properties of LogarithmsUse the properties of logarithms to verify that – ln ½ ln 2Rewriting Logarithmic ExpressionsExample 5: Rewriting the Logarithm of a Productlog105x3yExample 6: Rewriting the Logarithm of a Quotientln Example 7: Condensing a Logarithmic Expression log10x 3log10(x 1)Example 8: Condensing a Logarithmic Expression2ln (x 2) – ln x14

Section 3-4: Exponential and Logarithmic EquationsINTRODUCTIONIn this section, you will study procedures for solving equations involving exponential and logarithmicfunctions.A simple example:2 x 32This method does not work for an equation as simple as:ex 7The following properties are the INVERSE PROPERTIES of exponential and logarithmic functions.Base aBase e1.log a a x xlne x x2.a log a x xe ln x xSOLVING EXPONENTIAL AND LOGARITHMIC EQUATIONS1. To solve an exponential equation, first isolate the exponential expression, then take thelogarithms of both sides and solve for the variable.2. To solve for a logarithmic equation, rewrite the equation in exponential form and solve forthe variable.15

SOLVING EXPONENTIAL EQUATIONSExample 1: Solving an Exponential EquationSolvee x 72 .Example #2: Solving an Exponential EquationSolvee x 5 60 .Example #3: Solving an Exponential EquationSolve4e 2x 5 .Example #4: Solving an Exponential EquationSolvee 2x 3e x 2 0 .16

SOLVING LOGARITHMIC EQUATIONSTo solve a logarithmic equation such asln x 3Logarithmic formwrite the equation in exponential form as follows.e ln x e 3Exponentiate both sides.x e3Exponential formThe procedure is called exponentiating both sides of an equation.Example #5: Solving a Logarithmic EquationSolveln x 2 .Example #6: Solving a Logarithmic EquationSolve5 2ln x 4 .Example #7: Solving a Logarithmic EquationSolve2ln 3x 4.17

Example #8: Solving a Logarithmic EquationSolveln x ln(x 1) 1.In solving exponential or logarithmic equations, the following properties are useful. Can you seewhere these properties were used?1.x yif and only ifloga x log a y.2.x yif and only ifa x ay , a 0, a 1.Example #9: Doubling an InvestmentYou have deposited 500 in an account that pays 6.75% interest, compounded continuously. Howlong will it take your money to double?Example #10: Consumer Price Index for SugarFrom 1970 to 1993, the Consumer Price Index (CPI) value y for a fixed amount of sugar for the year tcan be modeled by the equationy 169.8 86.8ln twhere t 10 represents 1970. During which year did the price of sugar reach 4 times its 1970 price of30.5 on the CPI?18

Section 3-5: Exponential and Logarithmic ModelsINTRODUCTIONThe five most common types of mathematical models involving exponentialand logarithmic functions are as follows:1. Exponential growth model:y ae bx ,2. Exponential decay model:y ae bx ,3. Gaussian model:4. Logistics growth model:5. Logarithmic models:y aeb 0b 0 ( x b )2 cay 1 be ( x c ) dy a bln x and y a blog10 xThe graphs of the basic forms of these functions are as follows:y exy e x19

2 xy ey 1 ln xy 11 e xy 1 log x1020

EXPONENTIAL GROWTH AND DECAYExample 1: Population IncreaseEstimates of the world population (in millions) from 1980 to 1992 are shownin the table.Year1980Population4453(in 02451125202529453845478An exponential growth model that approximates this data is given byP 4451e 0.017303 t ,0 t 12where P is the population (in millions) and t 0 represents 1980. Accordingto this model, when will the world population reach 6 billion?Example #2: Finding an Exponential Growth ModelFind an exponential growth model whose graph passes through the points(0, 4453) and (7, 5024).21

In living organic material, the ratio of the number of radioactive carbonisotopes (carbon 14) to the number of nonradioactive carbon isotopes(carbon 12) is about 1 to 1012. When organic material dies, its carbon 12content remains fixed, whereas its radioactive carbon 14 begins to decaywith a half-life of about 5700 years. To estimate the age of dead organicmaterial, scientists use the following formula, which denotes the ratio ofcarbon 14 to carbon 12 present at any time t (in years).R 1 t 822312 e10Example #3: Carbon DatingThe ratio of carbon 14 to carbon 12 in a newly discovered fossil isR 11013 .Estimate the age of the fossil.22

GAUSSIAN MODELSGaussian models are commonly used in probability and statistics torepresent populations that are normally distributed. The graph of aGaussian model is called a bell-shaped curve.Example #4: SAT ScoresIn 1993, the Scholastic Aptitude Test (SAT) scores for males roughlyfollowed a normal distribution given byy 0.0026e (x 500)248,000,200 x 800where x is the SAT score for mathematics. Sketch the graph of this function.From the graph, estimate the average SAT score.23

LOGISTICS GROWTH MODELSExample #5: Spread of a VirusOn a college campus of 5000 students, one student returns from vacationwith a contagious flu virus. The spread of the virus is modeled by5000y 0.8t , 0 t1 4999ewhere y is the total number infected after t days. The college will cancelclasses when 40% or more of the students are ill.a) How many are infected after 5 days?b) After how many days will the college cancel classes?24