College Algebra Twelfth Edition Chapter 4 Inverse Exponential

Section 4. 3 Logarithmic Functions • Logarithmic Equations • Logarithmic Functions • Properties of

Logarithms (1 of 3) The previous section dealt with exponential functions of the form for all positive The horizontal line values of a, where test shows that exponential functions are one -to-one, and thus have inverse functions. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 3

Logarithms (2 of 3) The equation defining the inverse of a function is found by interchanging x and y in the equation that defines the function. Starting with and interchanging x and y yields Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 4

Logarithms (3 of 3) Here y is the exponent to which a must be raised in order to obtain x. We call this exponent a logarithm, symbolized by the abbreviation “log. ” The expression represents the logarithm in this discussion. The number a is called the base of the logarithm, and x is called the argument of the expression. It is read “logarithm with base a of x, ” or “logarithm of x with base a, ” or “base a logarithm of x. ” Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 5

Logarithm For all real numbers y and all positive numbers a and x, where if and only if The expression represents the exponent to which the base a must be raised in order to obtain x. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 6

Example 1: Writing Equivalent Logarithmic and Exponential Forms (1 of 2) The table shows several pairs of equivalent statements, written in both logarithmic and exponential forms. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 7

Example 1: Writing Equivalent Logarithmic and Exponential Forms (2 of 2) To remember the relationships among a, x, and y in the two equivalent forms refer to these diagrams. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 8

Example 2: Solving Logarithmic Equations (1 of 4) Solve each equation. (a) Solution Write in exponential form. Take cube roots Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 9

Example 2: Solving Logarithmic Equations (2 of 4) Check Original equation Let Write in exponential form. True The solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 10

Example 2: Solving Logarithmic Equations (3 of 4) Solve each equation. (b) Solution Write in exponential form. Apply the exponent. The solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 11

Example 2: Solving Logarithmic Equations (4 of 4) Solve each equation. (c) Solution Write in exponential form. Write with the same base. Power rule for exponents. Set exponents equal. Divide by 2. The solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 12

Logarithmic Function and x > 0, then the logarithmic If a > 0, function

Logarithmic Functions (1 of 3) Exponential and logarithmic functions are inverses of each other. The graph of is shown in red. The graph of its inverse is found by reflecting the graph of across the line Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 14

Logarithmic Functions (2 of 3) The graph of the inverse function, defined by shown in blue, has the y-axis as a vertical asymptote. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 15

Logarithmic Functions (3 of 3) The domain of an exponential function is the set of all real numbers, so the range of a logarithmic function also will be the set of all real numbers. In the same way, both the range of an exponential function and the domain of a logarithmic function are the set of all positive real numbers. Thus, logarithms can be found for positive numbers only. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 16

Logarithmic Function f of x equals log (base a) x (1 of 6) Domain: Range: For • is increasing and continuous on its entire domain, Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 17

Logarithmic Function f of x equals log (base a) x (2 of 6) Domain: Range: For • The y-axis is a vertical asymptote as from the right. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 18

Logarithmic Function f of x equals log (base a) x (3 of 6) Domain:

Logarithmic Function f of x equals log (base a) x (4 of 6) Range: Domain: For • for 0 < a < 1, is decreasing and continuous on its entire domain, Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 20

Logarithmic Function f of x equals log (base a) x (5 of 6) Domain: Range: For • The y-axis is a vertical asymptote as from the right. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 21

Logarithmic Function f of x equals log (base a) x (6 of 6) Domain:

Characteristics of the Graph of f of x equals log (base a) x 1. The points are on the graph. 2. If a > 1, then f is an increasing function. If 0 < a < 1, then f is a decreasing function. 3. The y-axis is a vertical asymptote. 4. The domain is and the range is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 23

Properties of Logarithms (1 of 5) The properties of logarithms enable us to change the form of logarithmic statements so that products can be converted to sums, quotients can be converted to differences, and powers can be converted to products. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 37

Properties of Logarithms (2 of 5) For x > 0, y > 0, and

Properties of Logarithms (3 of 5) For x > 0, y > 0, and

Properties of Logarithms (4 of 5) For x > 0, y > 0, and

Properties of Logarithms (5 of 5) For x > 0, y > 0, and

Example 5: Using the Properties of Logarithms (1 of 7) Rewrite each expression. Assume all variables represent positive real numbers, with (a) Solution Product property Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 42

Example 5: Using the Properties of Logarithms (2 of 7) Rewrite each expression. Assume all variables represent positive real numbers, with (b) Solution Quotient property Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 43

Example 5: Using the Properties of Logarithms (3 of 7) Rewrite each expression. Assume all variables represent positive real numbers, with (c) Solution Power property Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 44

Example 5: Using the Properties of Logarithms (4 of 7) Rewrite each expression. Assume all variables represent positive real numbers, with (d) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 45

Example 5: Using the Properties of Logarithms (5 of 7) Rewrite each expression. Assume all variables represent positive real numbers, with (e) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 46

Example 5: Using the Properties of Logarithms (6 of 7) Rewrite each expression. Assume all variables represent positive real numbers, with (f) Solution Power property Product and quotient properties Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 47

Example 5: Using the Properties of Logarithms (7 of 7) Rewrite each expression. Assume all variables represent positive real numbers, with (f) Solution Power property Distributive property Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 48

Example 6: Using the Properties of Logarithms (1 of 5) Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with (a) Solution Product and quotient properties Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 49

Example 6: Using the Properties of Logarithms (2 of 5) Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with (b) Solution Power property Quotient property Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 50

Example 6: Using the Properties of Logarithms (3 of 5) Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with (c) Solution Power property Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 51

Example 6: Using the Properties of Logarithms (4 of 5) Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with (c) Solution Product and quotient properties Rules for exponents Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 52

Example 6: Using the Properties of Logarithms (5 of 5) Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with (c) Solution Rules for exponents Definition of Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 53

Caution (2 of 2) There is no property of logarithms to rewrite a logarithm of a sum or difference. That is why, in Example 6(a), was not written The distributive property does not apply in a situation like this because is one term. The abbreviation “log” is a function name, not a factor. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 54

Example 7: Using the Properties of Logarithms with Numerical Values (1 of 2) Assume that without using a calculator. find each logarithm (a) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 55

Example 7: Using the Properties of Logarithms with Numerical Values (2 of 2) Assume that without using a calculator. find each logarithm (b) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 56

Theorem on Inverses (1 of 2) For a > 0, the following properties hold.

Theorem on Inverses (2 of 2) The following are examples of applications of this theorem. and The second statement in theorem will be useful when we solve other logarithmic and exponential equations. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 58