A P where A is Total amount, P is beginning amount, r is interest rate, t is time in years. You will need to use 2 nd ( ) ( )

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1 MATH 1314 College Algera Notes Spring 2012 Chapter 4: Exponential and Logarithmic Functions 1 Chapter 4.1: Exponential Functions x Exponential Functions are of the form f(x), where the ase is a numer 0 ut not equal to 1 and where x is any real numer. The x exponential function f(x) is read as the exponential function f with ase. Exponential Functions are very useful in real-world applications. They are used to model situations involving Inflation of Cost, Financial Investments, Population Growth, Growth of Epidemics, Radioactive Decay, and more. Ojective 1, 2: Evaluating and Graphing Exponential Functions(p413) Evaluating exponential functions means to plug in a known x-value into the function and calculate the result. You will need to use the ^ key on your TI-83/84 calculator. Use when evaluating functions. x Characteristics of Exponential Functions f(x) and their graphs: Domain is, 0, Range is 0 y-intercept is 1 ecause f(0) 1 (where 0) x if ase 1, then f(x) increases x if ase 0 < <1, then f(x) decreases is the equation for the Horizontal Asymptote. x (From the graph, f(x) is a one-to-one function, so it has an.) Ojective 3, 4: Natural ase e; Compound Interest Formulas(p417,419) 1 The irrational numer e is a value that 1 approaches as n n. Use a TABLE and enter large values of n,(10, 100, 1000, 10000,...) and n 1 you can see that the value 1 is approximately. n We call this approximate value the natural ase e. It is used to model natural exponential ehavior that increases or decreases. There are 2 Compound Interest Formulas used to calculate Total Investment amounts in this section: n ( n) ( t) r For n compounding periods per year use AP1 where n A is Total amount, P is eginning amount, r is interest rate, t is time in years, and n is numer of compounding periods per year. ( r)( ) For continuous compounding use e t A P where A is Total amount, P is eginning amount, r is interest rate, t is time in years. You will need to use 2 nd ( ) ( ) LN when entering e r t.

2 Practice prolems 4.1 Spring

3 Practice prolems 4.1 Spring

4 Practice prolems 4.1 Spring

5 MATH 1314 College Algera Notes Spring 2012 Chapter 4: Exponential and Logarithmic Functions 5 Chapter 4.2: Logarithmic Functions Logarithmic Functions are of the form f(x) log (x), where the ase is a numer ut not equal to 1 and where x 0. The functionf(x) log (x) is read as the logarithmic function f with ase. Logarithms are merely an exponent for an indicated ase. Logarithmic Functions are very useful in real-world applications and are used to model Earthquake and Sound Intensity, Acidity of Aqueous Solutions, Human Memory, and more. Ojective 1,2 3: Change etween Logarithmic and Exponential form; Evaluate logarithms(p425) To change etween exponential and logarithmic forms using the crossing method: line up the equal signs identify ase and write as exponential or logarithmic form switch x and y expressions y y log (x) logarithmic form x exponential form y x exponential form y log (x) logarithmic form To evaluate logarithms y inspection, rememer that the value of a logarithm is merely an for an indicated ase. Example: Evaluate log 2(32). () The value of log 2(32) is 5 ecause In logarithms such as log100, the lank ase is understood to e. This type of logarithm is called a Logarithm and can e log 100. You can use the LOG key to evaluate it. rewritten as 10 In logarithms such as ln(4), the notation ln represents log e and is called a Logarithm. You must use the LN key to evaluate it. Ojective 4: Basic Properties of Logarithms(p426) Recall that the value of a logarithm is merely an for the indicated ase. The following properties are ased on this fact. 1 log ( ) 1 ecause 0 log (1) 0 ecause 1 log ( x ) x log and ase in ( ) cancel leaving only x. log (x) x ase and log cancel out leaving only x.

MATH 1314 College Algera Notes Spring 2012 6 Chapter 4.

6 MATH 1314 College Algera Notes Spring Chapter 4.2: Logarithmic Functions Ojective 5,6: Graphing Logarithmic Functions(p427) Characteristics of Logarithmic Functions f(x) log (x) and their graphs: Domain is Range is x-intercept is if ase 1, then f(x) log (x) increases if ase 0 < <1, then f(x) log (x) decreases is the equation for the Asymptote. x Note: Since f(x) log (x) is one-to-one and is the inverse of f(x), the 1 that are on the graph of 1, 1, 1,0,,1 on the graph of f(x) log (x). points 1,, 0,1, 1, the points x f(x) will e reflected as Practice prolems 4.2 Spring 2012

7 Practice prolems 4.2 Spring

8 Practice prolems 4.2 Spring

9 MATH 1314 College Algera Notes Spring 2012 Chapter 4: Exponential and Logarithmic Functions 9 Chapter 4.3: Properties of Logarithms This section introduces properties of logarithms which will e used to rewrite ( or ) logarithmic expressions. When solving logarithmic equations or evaluating logarithmic expressions, it is sometimes necessary to rewrite logarithms using their properties. Certain characteristics of the logarithm properties will seem familiar ecause the properties of discussed previously correspond to properties of logarithms. Ojective 1,2,3,4,5,6: Properties of Logarithms; Change of Base Formula(p ) To expand logarithmic expressions, use the following order: Rule The expression log MN Rule The expression expands to log (M) log (N) M N log expands to log (M) log (N) p Rule The expression log M expands to p log M To condense logarithmic expressions, use the following order: p Rule The expression p log M condenses to log M Rule Rule The expression log (M) log (N) condenses to log MN The expression log (M) log (N) condenses to log M N The Change of Base Formula is used to evaluate a logarithm expression that has an indicated ase other than 10 or e. log(n) ln(n) To evaluate log (n) or log( ) ln( ). Example: Evaluate log 6(32). Round your answer to the nearest tenth. This logarithm uses ase 6. If your calculator does not have a LOG key for ase 6, then the Change of Base Formula is needed. Enter log 6(32) as log(32) or as ln(32) log(6) ln(6). Either entry gives. The answer is therefore.

10 Practice prolems 4.3 Spring

11 Practice prolems 4.3 Spring

12 MATH 1314 College Algera Notes Spring 2012 Chapter 4: Exponential and Logarithmic Functions 12 Chapter 4.4: Solving Exponential and Logarithmic Equations This section will now introduce methods for solving Exponential and Logarithmic Equations, including the TI-83/84 calculator. Ojective 1: Solving Exponential Equations(p448) To solve exponential equations algeraically using like ases: Make sure oth sides of equation have the. Rewrite if necessary. If ases cannot e made the same, use logarithms to solve. Once the ases are the same, and set the exponents equal to each other. Solve for x. M N To solve exponential equations using a TABLE: Enter left side of equation as Y1 and right side of equation as Y2. Press 2 nd GRAPH to find the solution for x in a TABLE. The solution will e the x-value with the same Y1 and Y2 value. M N To solve exponential equations using GRAPH: Enter left side of equation as Y1 and right side of equation as Y2. Press 2 nd TRACE to select the 5: intersect command. For First Curve? move cursor on first graph close to intersection and press ENTER. For Second Curve? move cursor on second graph close to intersection and press ENTER. Press ENTER again for Guess? IMPORTANT: If exact answers are needed, you may need to use the method aove. To solve exponential equations algeraically with logarithms: the exponential expression. Apply the logarithm or the logarithm to each side. Use the Rule for logarithms to ring the variale x expression down from the exponent and solve for x.

MATH 1314 College Algera Notes Spring 2012 13 Chapter 4.

13 MATH 1314 College Algera Notes Spring Chapter 4.4: Solving Exponential and Logarithmic Equations Ojective 2,3,4: Solving Logarithmic Equations(p449) For equations having logarithms on one side of equation only: the left side of equation to form a single logarithm. If the left side of equation only has a single logarithm, then you are ready for next step. Change the logarithm equation to form using the crossing method from section 4.2. Solve the new exponential equation using the like ases method. For equations containing logarithms on oth sides with the same ase: oth sides of equation until it is of the form log (M) log (N) with each side having a coefficient of 1. Using the one-to-one property, you can the log notation on oth sides and set the (M and N expressions) equal to each other. Solve for the variale in the new equation. To solve logarithmic equations with the TI-83/84 calculator, use a TABLE or GRAPH y following the same steps given earlier for exponential equations. Practice prolems 4.4 Spring 2012

14 Practice prolems 4.4 Spring

15 Practice prolems 4.4 Spring

16 16 MATH 1314 College Algera Notes Spring 2012 Chapter 4: Exponential and Logarithmic Functions Chapter 4.5: Exponential Growth and Decay In this section, you will learn how to create functions to model exponential growth and exponential decay and use them to make future predictions. Ojective 1: Exponential Growth and Decay(p460) Exponential Growth and Exponential Decay use the same mathematical model(you will use this model to make a formula): ( k)( ) A A e t 0 where A is the final amount of a sample at time t A is the eginning amount of a sample when time t = 0 0 k is the growth rate if k > 0 or is the decay rate if k < 0 (This constant represents the percentage of increase or decrease in the population or sample) t is time Important terms to know for Growth: increase eginning amount grows doule2 times eginning amount triple3 times eginning amount Important terms to know for Decay: decrease eginning amount reaks down, decomposes half-lifetime needed for 1 of a sustance to decay. 2 Ojective 2: Logistic Growth(p464) In real life, exponential growth is limited y conditions set y nature, therefore, population growth will e limited y things like the surrounding resources availale and the environment. An epidemic will grow and egin to spread exponentially within a confined population, ut will eventually slow down as the numer of people affected approaches the population size. Limited Logistic Growth for populations uses the mathematical model: c A ( )( ) 1 e t a where A is the size of the population affected at time t c is the limiting(maximum) size of A as time t a and are constants t is time

17 Practice prolems 4.5 Spring

18 Practice prolems 4.5 Spring

19 Practice prolems 4.5 Spring

Unit: Logarithms (Logs)

Unit: Logarithms (Logs) NAME Per http://www.mathsisfun.com/algera/logarithms.html /8 pep rally Introduction of Logs HW: Selection from Part 1 /1 ELA A.11A Introduction & Properties of Logs (changing forms)