Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Logarithmic Function Learn to define logarithmic functions. Learn to evaluate logarithms. Learn to find the domains of logarithmic functions. Learn to graph logarithmic functions. Learn to use logarithms to evaluate exponential equations. SECTION

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Converting from Exponential to Logarithmic Form Write each exponential equation in logarithmic form. Solution

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Converting from Logarithmic Form to Exponential Form Write each logarithmic equation in exponential form. Solution

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Evaluating Logarithms Find the value of each of the following logarithms. Solution

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Evaluating Logarithms Solution continued

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Using the Definition of Logarithm Solve each equation. Solution

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Using the Definition of Logarithm Solution continued

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Using the Definition of Logarithm Solution continued

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DOMAIN OF LOGARITHMIC FUNCTION Domain of f –1 (x) = log a x is (0, ∞) Range of f –1 (x) = log a x is (–∞, ∞) Recall that Domain of f (x) = a x is (–∞, ∞) Range of f (x) = a x is (0, ∞) Since the logarithmic function is the inverse of the exponential function, Thus, the logarithms of 0 and negative numbers are not defined.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Finding the Domain Find the domain of each function. Solution a.Domain of a logarithmic function must be positive, that is, The domain of f is (–∞, 2).

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Finding the Domain Solution continued b.Domain must be positive, that is, The domain of f is (–∞, –1) U (2, ∞). Set numerator = 0 and denominator = 0. x – 2 = 0 x + 1 = 0 x = 2 x = –1 Create a sign graph.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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,

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Sketching a Graph Sketch the graph of y = log 3 x. Solution by plotting points. Make a table of values. xy = log 3 x(x, y) 3 –3 = 1/27–3(1/27, –3) 3 –2 = 1/9–2(1/9, –2) 3 –3 = 1/3–1(1/3, –1) 3 0 = 10(1, 0) 3 1 = 31(3, 1) 3 2 = 92(9, 2)

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Sketching a Graph Solution by using the inverse function 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

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PROPERTIES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS Exponential Function f (x) = a x Logarithmic Function f (x) = log a x Domain (0, ∞) Range (–∞, ∞) Domain (–∞, ∞) Range (0, ∞) x-intercept is 1 No y-intercept y-intercept is 1 No x-intercept x-axis (y = 0) is the horizontal asymptote y-axis (x = 0) is the vertical asymptote

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 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

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley GRAPHS OF LOGARITHMIC FUNCTIONS f (x) = log a x (0 < a < 1) f (x) = log a x (a > 1)

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Using Transformations Solution Shift up 2 Domain (0, ∞) Range (–∞, ∞) Vertical asymptote x = 0

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Using Transformations Solution continued Shift right 1 Domain (1, ∞) Range (–∞, ∞) Vertical asymptote x = 1

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Using Transformations Solution continued Reflect graph of y = log 3 x in the x-axis Domain (0, ∞) Range (–∞, ∞) Vertical asymptote x = 0

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Using Transformations Solution continued Reflect graph of y = log 3 x in the y-axis Domain (∞, 0) Range (–∞, ∞) Vertical asymptote x = 0

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Using Transformations to Sketch a Graph Sketch the graph of Solution Graph f (x) = log x and shift it right 2 units.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Using Transformations to Sketch a Graph Solution continued Reflect in x-axisShift up 2

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Evaluating the Natural Logarithm Evaluate each expression. Solution (Use a calculator.)

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 10 McDonald’s Hot Coffee The local McDonald’s franchise has discovered that when coffee is poured from a pot whose contents are at 180ºF into a noninsulated pot in the store where the air temperature is 72ºF, after 1 minute the coffee has cooled to 165ºF. How long should the employees wait before pouring the coffee from this noninsulated pot into cups to deliver it to customers at 125ºF?

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 10 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.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 10 McDonald’s Hot Coffee Substitute this value for k. Solution continued Solve for t when T = 125. The employee should wait about 5 minutes.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 11 Chemical Toxins in a Lake In a large lake, one-fifth of the water is replaced by clean water each year. A chemical spill deposits 60,000 cubic meters of soluble toxic waste into the lake. a.How much of this toxin will be left in the lake after four years? b.How long will it take for the toxic chemical in the lake to be reduced to 6000 cubic meters?

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 11 Chemical Toxins in a Lake One-fifth (1/5) of the water in the lake is replaced by clean water every year, the decay rate for the toxin is r = –1/5 and A 0 = 60,000. So, Solution where A is the amount of toxin (in cubic meters) after t years.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 11 Chemical Toxins in a Lake a. Substitute t = 4. Solution continued b. Substitute A = 6000 and solve for t.