Precalculus – MAT 129 Instructor: Rachel Graham Location: BETTS Rm. 107 Time: 8 – 11:20 a.m. MWF

Chapter Three Exponential and Logarithmic Functions

Ch. 3 Overview Exponential Fxns and Their Graphs Logarithmic Fxns and Their Graphs Properties of Logarithms Solving Exponential and Logarithmic Equations Exponential and Logarithmic Models Nonlinear Models

3.1 – Exponential Fxns and Their Graphs Exponential Functions Graphs of Exponential Functions The Natural Base e Applications

3.1 – Exponential Functions The exponential function f with base a is denoted by: f(x)=a x

3.1 – Graphs of Exponential Fxns Figure 3.1 on pg. 185 shows the form of the graph of: y=a x Figure 3.2 on pg. 185 shows the form of the graph of: y=a -x

Example Pg. 187 Example 4 After looking at the solution read the paragraph at the bottom of the page.

3.1 – The Natural Base e e≈ –Useful for a base in many situations. f(x)=e x is called the natural exponential function.

Example Pg. 189 Example 6 Be sure you know how to evaluate this function on your calculator.

3.1 – Applications The most widely used application of the exponential function is for showing investment earnings with continuously compounded interest.

Formulas for Compounding Interest After 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+r/n) nt 2.For continuous compounding: A=Pe rt

Example Pg. 191 Examples 8 and 9. You will be responsible for knowing the compound interest formula.

Activities (191) 1. Determine the balance A at the end of 20 years if $1500 is invested at 6.5% interest and the interest is compounded (a) quarterly and (b) continuously. 2. Determine the amount of money that should be invested at 9% interest, compounded monthly, to produce a final balance of $30,000 in 15 years.

3.2 – Logarithmic Fxns and Their Graphs Logarithmic Functions Graphs of Logarithmic Functions The Natural Logarithmic Function Applications

3.2 – Logarithmic Functions The inverse of the exponential function is the logarithmic function. For x>0, a>0, and a≠1, y=log a x if and only if x=a y. f(x)=log a x is called the logarithmic function with base a.

Properties of Logarithms 1.log a 1=0 because a 0 =1. 2.log a a=1 because a 1 =a. 3.log a a x =x because a logx =x. 4.If log a x=log a y, then x=y

Example Pg. 203 #33. Solve the equation for x. log 7 x=log 7 9

Solution Example Pg. 203 #33. x=9

3.2 – Graphs of Logarithmic Fxns See beige box on pg. 199

3.2 – The Natural Logarithmic Fxn For x>0, y=ln x if and only if x=e y. f(x) = log e x = ln x is called the natural logarithmic function.

Properties of Natural Logarithms 1.ln 1=0 because e 0 =1. 2.ln e=1 because e 1 =e. 3.ln e x =x because e ln x =x. 4.If ln x=ln y, then x=y

Example Pg. 201 Example 9. Note both the algebraic and graphical solutions.

3.2 – Application See example 10 on pg. 202 for the best application of logarithmic functions.

3.3 – Properties of Logarithms Change of Base Properties of Logarithms Rewriting Logarithmic Expressions

3.3 – Change of Base To evaluate logarithms at different bases you can use the change of base formula: log a x = (log b x/ log b a)

Example Pg. 207 Examples 1 & 2. Note both log and ln functions will yield the same result.

3.3 – Properties of Logarithms See blue box on pg. 208.

Example Pg. 208 Example 3 These should be pretty self explanatory.

3.3 – Rewriting Log Fxns This is where you use the multiplication, division, and power rules to expand and condense logarithmic expressions.

Example Pg. 209 Examples 5&6. Note that a square root is equal to the power of ½.

3.4 – Solving Exponential and Logarithmic Equations Introduction Solving Exponential Equations Solving Logarithmic Equations Applications

3.5 –Exponential and Logarithmic Models Introduction Exponential Growth and Decay Gaussian Models Logistic Growth Models Logarithmic Models

The Models

Example Example 2 on pg. 227 In a research experiment, a population of fruit flies is increasing according to the law of exponential growth. After 2 days there are 100 fruit flies, and after 4 days there are 300 fruit flies. How many flies will there be after 5 days?

Example Example 5 on pg. 230 On a college campus of 5000 students, one student returns from vacation with a contagious flu virus. The spread of the virus is modeled on pg. 230 where y is the total number infected after t days. The college will cancel classes when 40% or more are infected. a)How many students are infected after 5 days? b)After how many days will the college cancel classes?

Example On the Richter scale, the magnitude R of an earthquake of intensity I is given by R = log 10 I/I 0 where I 0 = 1 is the minimum intensity used for comparison. Intensity is a measure of wave energy of an earthquake.

Activities In Class QUIZ: pp. 234 #30, 41a, 42a.