2. Copyright © 2007 Pearson Education, Inc. Slide 5-2 Chapter 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions 5.3 Logarithms and Their Properties 5.4 Logarithmic Functions 5.5 Exponential and Logarithmic Equations and Inequalities 5.6 Further Applications and Modeling with Exponential and Logarithmic Functions

3. Copyright © 2007 Pearson Education, Inc. Slide 5-3 5.3 Logarithms and Their Properties Logarithm For all positive numbers a, where a  1, A logarithm is an exponent, and log a x is the exponent to which a must be raised in order to obtain x. The number a is called the base of the logarithm, and x is called the argument of the expression log a x. The value of x will always be positive.

4. Copyright © 2007 Pearson Education, Inc. Slide 5-4 5.3 Examples of Logarithms Exponential Form Logarithmic Form Example Solve Solution

5. Copyright © 2007 Pearson Education, Inc. Slide 5-5 5.3 Solving Logarithmic Equations ExampleSolve a) Solution a) Since the base must be positive, x = 2.

6. Copyright © 2007 Pearson Education, Inc. Slide 5-6 5.3 The Common Logarithm – Base 10 ExampleEvaluate SolutionUse a calculator. For all positive numbers x,

7. Copyright © 2007 Pearson Education, Inc. Slide 5-7 ExampleIn chemistry, the pH of a solution is defined as where [H 3 O + ] is the hydronium ion concentration in moles per liter. The pH value is a measure of acidity or alkalinity of a solution. Pure water has a pH of 7.0, substances with a pH greater than 7.0 are alkaline, and those less than 7.0 are acidic. a)Find the pH of a solution with [H 3 O + ] = 2.5×10 -4. b)Find the hydronium ion concentration of a solution with pH = 7.1. Solution a)pH = –log [H 3 O + ] = –log [2.5×10 -4 ]  3.6 b)7.1 = –log [H 3 O + ]  –7.1 = log [H 3 O + ]  [H 3 O + ] = 10 -7.1  7.9 ×10 -8 5.3 Application of the Common Logarithm

8. Copyright © 2007 Pearson Education, Inc. Slide 5-8 5.3 The Natural Logarithm – Base e On the calculator, the natural logarithm key is usually found in conjunction with the e x key. For all positive numbers x,

9. Copyright © 2007 Pearson Education, Inc. Slide 5-9 5.3 The Graph of ln x and Some Calculator Examples ExampleEvaluate Solution

10. Copyright © 2007 Pearson Education, Inc. Slide 5-10 5.3 Using Natural Logarithms to Solve a Continuous Compounding Problem Example Suppose that $1000 is invested at 3% annual interest, compounded continuously. How long will it take for the amount to grow to $1500? Analytic Solution

11. Copyright © 2007 Pearson Education, Inc. Slide 5-11 5.3 Using Natural Logarithms to Solve a Continuous Compounding Problem Graphing Calculator Solution Let Y 1 = 1000e.03t and Y 2 = 1500. The table shows that when time (X) is 13.5 years, the amount (Y 1 ) is 1499.3  1500.

12. Copyright © 2007 Pearson Education, Inc. Slide 5-12 5.3 Properties of Logarithms Property 1 is true because a 0 = 1 for any value of a. Property 2 is true since in exponential form: Property 3 is true since log a k is the exponent to which a must be raised in order to obtain k. For a > 0, a  1, and any real number k, 1.log a 1 = 0, 2.log a a k = k, 3.a log a k = k, k > 0.

13. Copyright © 2007 Pearson Education, Inc. Slide 5-13 5.3 Additional Properties of Logarithms Examples Assume all variables are positive. Rewrite each expression using the properties of logarithms. 1. 2. 3. For x > 0, y > 0, a > 0, a  1, and any real number r, Product Rule Quotient Rule Power Rule

14. Copyright © 2007 Pearson Education, Inc. Slide 5-14 5.3 Example Using Logarithm Properties Example Assume all variables are positive. Use the properties of logarithms to rewrite the expression Solution

15. Copyright © 2007 Pearson Education, Inc. Slide 5-15 5.3 Example Using Logarithm Properties Example Use the properties of logarithms to write as a single logarithm with coefficient 1. Solution

16. Copyright © 2007 Pearson Education, Inc. Slide 5-16 5.3 The Change-of-Base Rule ProofLet Change-of-Base Rule For any positive real numbers x, a, and b, where a  1 and b  1,

17. Copyright © 2007 Pearson Education, Inc. Slide 5-17 5.3 Using the Change-of-Base Rule Example Evaluate each expression and round to four decimal places. (a) Solution Note in the figures below that using either natural or common logarithms produce the same results.

18. Copyright © 2007 Pearson Education, Inc. Slide 5-18 5.3 Modeling the Diversity of Species Example One measure of the diversity of species in an ecological community is the index of diversity, where and P 1, P 2,..., P n are the proportions of a sample belonging to each of n species found in the sample. Find the index of diversity in a community where there are two species, with 90 of one species and 10 of the other.

19. Copyright © 2007 Pearson Education, Inc. Slide 5-19 5.3 Modeling the Diversity of Species Solution Since there are a total of 100 members in the community, P 1 = 90/100 =.9, and P 2 = 10/100 =.1. Interpretation of this index varies. If two species are equally distributed, the measure of diversity is 1. If there is little diversity, H is close to 0. In this case H .5, so there is neither great nor little diversity.