1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 4 Exponential and Logarithmic Functions

OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 Logarithmic Functions Define logarithmic functions. Evaluate logarithms. Find the domains of logarithmic functions. Graph logarithmic functions. Use logarithms to evaluate exponential equations. SECTION

3 © 2010 Pearson Education, Inc. All rights reserved DEFINITION OF THE LOGARITHMIC FUNCTION For x > 0, a > 0, and a ≠ 1, The function f (x) = log a x, is called the logarithmic function with base a. The logarithmic function is the inverse function of the exponential function.

4 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Converting from Exponential to Logarithmic Form Write each exponential equation in logarithmic form. Solution

5 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Converting from Logarithmic Form to Exponential Form Write each logarithmic equation in exponential form. Solution

6 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Evaluating Logarithms Find the value of each of the following logarithms. Solution

7 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 Evaluating Logarithms Solution continued

8 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Using the Definition of Logarithm Solve each equation. Solution

9 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Using the Definition of Logarithm Solution continued

10 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Using the Definition of Logarithm Solution continued

11 © 2010 Pearson Education, Inc. All rights reserved DOMAIN OF LOGARITHMIC FUNCTION Domain of y = log a x is (0, ∞) Range of y = log a x is (–∞, ∞) Logarithms of 0 and negative numbers are not defined.

12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Finding the Domain Find the domain of Solution Domain of a logarithmic function must be positive, that is, The domain of f is (–∞, 2).

13 © 2010 Pearson Education, Inc. All rights reserved BASIC PROPERTIES OF LOGARITHMS 1.log a a = 1. 2.log a 1 = 0. 3.log a a x = x, for any real number x. For any base a > 0, with a ≠ 1,

14 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Sketching a Graph Sketch the graph of y = log 3 x. Solution by plotting points (Method 1) Make a table of values.

15 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Sketching a Graph Solution continued Plot the ordered pairs and connect with a smooth curve to obtain the graph of y = log 3 x.

16 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Sketching a Graph Solution by using the inverse function (Method 2) Graph y = f (x) = 3 x. Reflect the graph of y = 3 x in the line y = x to obtain the graph of y = f –1 (x) = log 3 x. © 2010 Pearson Education, Inc. All rights reserved 16

17 © 2010 Pearson Education, Inc. All rights reserved GRAPHS OF LOGARITHMIC FUNCTIONS

18 © 2010 Pearson Education, Inc. All rights reserved PROPERTIES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS Exponential Function f (x) = a x Logarithmic Function f (x) = log a x Domain (0, ∞) Range (–∞, ∞) 1.Domain (–∞, ∞) Range (0, ∞) x-intercept is 1 No y-intercept 2. y-intercept is 1 No x-intercept 3. x-axis (y = 0) is the horizontal asymptote y-axis (x = 0) is the vertical asymptote

19 © 2010 Pearson Education, Inc. All rights reserved PROPERTIES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS Exponential Function f (x) = a x Logarithmic Function f (x) = log a x The graph is a continuous smooth curve that passes through the points (1, 0), and (a, 1). 4.The graph is a continuous smooth curve that passes through the points (0, 1), and (1, a).

20 © 2010 Pearson Education, Inc. All rights reserved PROPERTIES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS Exponential Function f (x) = a x Logarithmic Function f (x) = log a x Is one-to-one, that is, log a u = log a v if and only if u = v. 5. Is one-to-one, that is, a u = a v if and only if u = v. Increasing if a > 1 Decreasing if 0 < a < 1 6.Increasing if a > 1 Decreasing if 0 < a < 1

21 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Using Transformations Start with the graph of f (x) = log 3 x and use transformations to sketch the graph of each function. State the domain and range and the vertical asymptote for the graph of each function.

22 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Using Transformations Solution Shift up 2 Domain (0, ∞) Range (–∞, ∞) Vertical asymptote x = 0

23 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Using Transformations Solution continued Shift right 1 Domain (1, ∞) Range (–∞, ∞) Vertical asymptote x = 1

24 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Using Transformations Solution continued Reflect graph of y = log 3 x in the x-axis Domain (0, ∞) Range (–∞, ∞) Vertical asymptote x = 0

25 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Using Transformations Solution continued Reflect graph of y = log 3 x in the y-axis Domain (∞, 0) Range (–∞, ∞) Vertical asymptote x = 0

26 © 2010 Pearson Education, Inc. All rights reserved COMMON LOGARITHMS 1.log 10 = 1 2.log 1 = 0 3.log 10 x = x The logarithm with base 10 is called the common logarithm and is denoted by omitting the base: log x = log 10 x. Thus, y = log x if and only if x = 10 y. Applying the basic properties of logarithms

27 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Using Transformations to Sketch a Graph Sketch the graph of Solution Start with the graph of f (x) = log x. Step 1: Replacing x with x – 2 shifts the graph two units right.

28 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 8 Using Transformations to Sketch a Graph Solution continued Step 2: Multiplying by  1 reflects the graph Step 3: Adding 2 shifts the graph two units up. in the x-axis.

29 © 2010 Pearson Education, Inc. All rights reserved NATURAL LOGARITHMS 1.ln e = 1 2.ln 1 = 0 3.log e x = x The logarithm with base e is called the natural logarithm and is denoted by ln x. That is, ln x = log e x. Thus, y = ln x if and only if x = e y. Applying the basic properties of logarithms

30 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 9 Evaluating the Natural Logarithm Evaluate each expression. Solution Use a calculator.

31 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 10 Doubling Your Money a.How long will it take to double your money if it earns 6.5% compounded continuously? b. At what rate of return, compounded continuously, would your money double in 5 years? Solution a.If P is the original amount invested, A = 2P. It will take 11 years to double your money.

32 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 10 Doubling Your Money Solution continued b. Your investment will double in 5 years at the rate of 13.86%.

33 © 2010 Pearson Education, Inc. All rights reserved NEWTON’S LAW OF COOLING Newton’s Law of Cooling states that where T is the temperature of the object at time t, T s is the surrounding temperature, and T 0 is the value of T at t = 0.

34 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 11 McDonald’s Hot Coffee The local McDonald’s franchise has discovered that when coffee is poured from a coffeemaker whose contents are 180ºF into a noninsulated pot, after 1 minute, the coffee cools to 165ºF if the room temperature is 72ºF. How long should the employees wait before pouring the coffee from this noninsulated pot into cups to deliver it to customers at 125ºF?

35 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 11 McDonald’s Hot Coffee Use Newton’s Law of Cooling with T 0 = 180 and T s = 72 to obtain Solution We have T = 165 and t = 1.

36 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 11 McDonald’s Hot Coffee Substitute this value for k. Solution continued Solve for t when T = 125. The employee should wait about 5 minutes.

37 © 2010 Pearson Education, Inc. All rights reserved GROWTH AND DECAY MODEL A is the quantity after time t. A 0 is the initial (original) quantity (when t = 0). r is the growth or decay rate per period. t is the time elapsed from t = 0.

38 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 12 Chemical Toxins in a Lake A chemical spill deposits 60,000 cubic meters of soluble toxic waste into a large lake. If 20% of the waste is removed every year, how many years will it take to reduce the toxin to 1000 cubic meters? Solution In the equation A = A 0 e rt, we need to find A 0, r, and the time when A = 1000.

39 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 12 Chemical Toxins in a Lake 1.Find A 0. Initially (t = 0), we are given A 0 = 60,000. So Solution continued 2.Find r. When t = 1 year, the amount of toxin will be 80% of its initial value, or

40 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 12 Chemical Toxins in a Lake Solution continued 2.continued So

41 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 12 Chemical Toxins in a Lake Solution continued 3. Find t when A = It will take approximately 18 years to reduce toxin to 1000 m 3.