CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic Functions and Graphs 5.4 Properties of Logarithmic Functions 5.5 Solving Exponential and Logarithmic Equations 5.6 Applications and Models: Growth and Decay; and Compound Interest Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
5.4 Properties of Logarithmic Functions Convert from logarithms of products, powers, and quotients to expressions in terms of individual logarithms, and conversely. Simplify expressions of the type logaax and . Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Logarithms of Products The Product Rule For any positive numbers M and N and any logarithmic base a, loga MN = loga M + loga N. (The logarithm of a product is the sum of the logarithms of the factors.) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Express as a single logarithm: Solution: Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Logarithms of Powers The Power Rule For any positive number M, any logarithmic base a, and any real number p, (The logarithm of a power of M is the exponent times the logarithm of M.) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Examples Express as a product. Solution: Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Logarithms of Quotients The Quotient Rule For any positive numbers M and N, and any logarithmic base a, (The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Examples Express as a difference of logarithms: Solution: Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Examples Express as a single logarithm: Solution: Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Applying the Properties - Example Express each of the following in terms of sums and differences of logarithms. Solution: Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) Solution: Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) Solution: Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Express as a single logarithm: Solution: Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Examples Given that loga 2 ≈ 0.301 and loga 3 ≈ 0.477, find each of the following, if possible. Solution: Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Examples (continued) Solution: Cannot be found using these properties and the given information. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Expressions of the Type loga ax The Logarithm of a Base to a Power For any base a and any real number x, loga ax = x. (The logarithm, base a, of a to a power is the power.) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Examples Simplify. a) loga a8 b) ln et c) log 103k Solution: a. loga a8 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Expressions of the Type A Base to a Logarithmic Power For any base a and any positive real number x, (The number a raised to the power loga x is x.) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Examples Simplify. Solution: Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley