PreCalculus Notes MAT 129 Chapter 6: Exponential and Logarithmic - - PowerPoint PPT Presentation

PreCalculus Notes

MAT 129 Chapter 6: Exponential and Logarithmic Functions David J. Gisch

Department of Mathematics Des Moines Area Community College

September 30, 2011

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Chapter 6 Section 6.1: Composite Functions Section 6.2: One-to-One Functions; Inverse Functions Section 6.3: Exponential Functions Section 6.4: Logarithmic Functions Section 6.5: Properties of Logarithms Section 6.6: Logarithmic and Exponential Equations Section 6.7: Compound Interest Section 6.8: Exponential Growth and Decay; Newton’s Law; Etc Section 6.9: Building Exponential, Logarithmic, and Logistic Models from Data

Section 6.1: Composite Functions

Summary We will what composite functions are and how to find their domain and range.

Definition Given two functions f and g, the composite function, denoted by f ◦ g (read as “f composed of g” or “f of g”), is defined by (f ◦ g)(x) = f (g(x)) The domain of f ◦ g is the set of all numbers x in the domain of g such that g(x) is in the domain of f .

Example Let f (x) = 2x2 − 3 and g(x) = 3x. Find: (a) (f ◦ g)(1) (b) (f ◦ g)(3) (c) (f ◦ g)(−2)

Example Use the given functions to write one composite function. (a) Let f (x) = 2x2 − 3 and g(x) = 3x; (f ◦ g)(x). (b) Let f (x) = 2x2 − 3 and g(x) = 3x; (g ◦ f )(x). (c) Let f (x) = x−4

x2+1 and g(x) = 3x; (f ◦ g)(x).

Finding the Domain of (f ◦ g)(x)

1

You first exclude any elements that are not in the domain of g.

2

You find elements not in the domain of f and set g(x) equal to those values and solve. The resulting solutions are values thet are also excluded from the domain.

Find the Domain

Example (a) Let f (x) = 2x2 − 3 and g(x) = 3x.Find the domain of (f ◦ g)(x) (b) Let f (x) = 2x2 − 3 and g(x) = 1

x . Find the domain of (f ◦ g)(x)

(c) Let f (x) = √x − 6 and g(x) =

2 x−1. Find the domain of (f ◦ g)(x)

Example Given: f (x) = √x − 4, g(x) = x2 + 1, and h(x) = x−1

x+1, find each of the

following and state the domain. (a) (f ◦ g)(x) (b) (h ◦ g)(x) (c) (g ◦ f )(x)

Section 6.2: One-to-One Functions; Inverse Functions

Summary We will find out what it means for a function to be one-to-one; find inverse functions by analyzing graphs and equations; and graph inverse functions.

Definition A function is one-to-one if for any two inputs in the domain they map to two different outputs in the range. That is, if x1 = x2 are in the domain then f (x1) = f (x2).

Theorem When looking at the graph of a functions f , if every horizontal line intersects the graph in at most one point then f is one-to-one.

Theorem If f is one-to-one, then f is said to have an inverse f −1 such that if f (x) = y then f −1(y) = x. Therefore, if you know that the graph of f (a 1-1 function) contains the point (2, −3), then you know f −1 contains the point (−3, 2).

Theorem The graph of f and f −1 are symmetric with respect to the line y = x. Example Given the graph, graph the inverse.

Theorem Given f and its inverse f −1 then for all values x we have (f ◦ f −1)(x) = f (f −1(x)) = x and (f −1 ◦ f )(x) = f −1(f (x)) = x Example Check if f (x) = 3x − 4 and g(x) = 1

3x + 4 3 are inverses.

Calculating Inverses

To calculate the inverse of a function you swap the variable y and x and solve for y. Example Find the inverse of f (x) = 2x − 10.

Calculating Inverses

Example Find the inverse of f (x) = 2x2 − 10.

Calculating Inverses

Example Find the inverse of f (x) = 2x2 − 10 if x ≥ 0.

Calculating Inverses

Example Find the inverse of f (x) = 2x+1

x+1 .

Section 6.3: Exponential Functions

Summary We will evaluate exponential functions; graph them; and solve exponential equations.

Theorem Recall your laws of exponents. aman = am+n

am an = am−n

(an)m = am·n 1m = 1 a0 = 1 a−m =

1 am

Definition An exponential functions is a function of the form f (x) = ax where a is a positive real number. The domain of f is the set of all real numbers.

Theorem For an exponential function of the form f (x) = ax where a > 0, and a = 1, then f (x + 1) f (x) = a Example Let’s try it for f (x) = 4x.

Graphs of Exponential Functions

http://www.analyzemath.com/expfunction/expfunction.html Click here.

f (x) = ax, with a > 1

f (x) = ax, with 0 < a < 1

Example Graph f (x) = 2−x − 3 using transformations.

The Number e

Definition The number e is defined as the number such that e = lim

n→∞

• 1 + 1

n n

To see what we mean lets evaluate this for a few values of n. n 1 2 10 100 100,000 1,000,000 (1 + 1

n)n

2 2.25 2.593742 2.704814 2.718268 2.718280

The Graph of e

Figure: Graph of y = ex

Solving Exponential Equations

Example Solve 22x−1 = 32 Example Solve e2x−1 =

1 e3x ·

• e−x4

Solving Exponential Equations

Example

Section 6.4: Logarithmic Functions

Summary We will change exponential functions to logarithmic functions and vice versa; evaluate logarithmic functions; determine the domaina nd range; and solve logarithmic functions.

Definition A logarithmic function to the base a, where a > 0 and a = 1, is denoted by y = loga x (read as “log base a of x”) and is defined by y = loga x if and only if x = ay The domain of a logarithmic function of this form is x > 0.

Example Change to the alternate form. (a) For example, 34 = 81, so 4 = log3 81. (b) y = log3 x (c) y = log5 x (d) y = log2 32

Example Change to the alternate form. (a) 1.23 = m (b) eb = 9 (c) −3 = loge x

Domain of a Logarithmic Function

Note: The reason this is true is because the logarithmic function y = loga x is defined as the inverse function of y = ax. For any inverse f −1(x) the domain is the range of f (x). In this case f (x) is an exponential function which has the range of (0, ∞). Domain Range f (x) = ax (−∞, ∞) (0, ∞) f −1(x) = loga x (0, ∞) (−∞, ∞)

Example Find the domain of each logarithmic function. (a) f (x) = log3(x − 2) (b) g(x) = log2 x+3

x−1

• (c) h(x) = log 1

2 x2

Graphs of Logarithmic Functions

Recall that if f (x) contains the point (a, b), then f −1(x) contains the point (b, a). Thus, as logarithmic functions are inverses of exponential functions we can reflect (about y = x) the graph an exponential function to create the graph of a logarithmic function.

Graphs of Logarithmic Functions

The Natural Logarithm Function

Rather than write loge we have a special term for this and we call it the natural logarithm. y = ln x if and only if x = ey

Example (a) Find the domain of the logarithmic function f (x) = − ln(x − 3). (b) Graph f (x). (c) Determine the range and vertical asymptote. (d) Find f −1(x). (e) Use f −1(x) to find the range of f . (f) graph f −1(x).

Note: If we have a logarithm of base 10 we call this the common logarithm. In this case we do not write the number of the base. y = log x if and only if x = 10y

Example (a) Find the domain of the logarithmic function f (x) = 3 log(x + 5). (b) Graph f (x). (c) Determine the range and vertical asymptote. (d) Find f −1(x). (e) Use f −1(x) to find the range of f . (f) graph f −1(x).

Solving Logarithmic Equations

Example Solve each of the following (a) log3(4x − 7) = 2 (b) log2(2x + 1) = 3 (c) e2x+5 = 8 (d) ln(x + 2) = 5

Example The formula D = 5e−0.4h can be used to find the number of milligrams D of a certain drug that is in a patient’s bloodstream h hours after the drug was administered. When the number of milligrams reaches 2, the drug is to be administered again. What is the time between injections?

Section 6.5: Properties of Logarithms

Summary We will work with the properties of logarithms to rewrite logarithms in different forms and to evaluate logarithms whose base is neither 10 nor e.

Properties of Logarithms loga 1 = (1) loga a = 1 (2) aloga M = M (3) loga ar = r (4) loga(MN) = loga M + loga N (5) loga M N = loga M − loga N (6) loga Mr = r loga M (7)

Example Write log2(x2 3 √x − 1), with x > 1, as the sum of logarithms.

Example Write log6

x4 (x2+3)2 , with x = 0, as the difference of logarithms.

Example Write ln x3√x−2

(x+1)2 , with x > 2, as the sum and difference of logarithms.

Example Write 3 ln 2 + ln(x2) + 2 as a single logarithm.

Example Write 1

2 loga 4 − 2 loga 5 as a single logarithm.

Change of Base Formula

Theorem If a = 1, b = 1 and M are positive real numbers, then loga M = logb M logb a Example Evaluate log4 9 on your calculator.

Section 6.6: Logarithmic and Exponential Equations

Summary We will work with the properties of logarithms and exponents to solve equations.

Example Solve the following: 2 log4 x = log4 9

Example Solve the following: log2(x + 2) = log2(1 − x) = 1

Example Solve the following: 3x = 7

Example Solve the following: 5 · 2x = 3

Example Solve the following: 2x−1 = 52x+3

Example Solve the following: 5x−2 = 33x+2

Example Solve the following: 4x − 2x − 12 = 0

Example Solve the following: x + ex = 2

Example The population of the world in 2006 was 6.53 billion people and was growing at a rate of 1.14% per year. Assuming that this growth rate continues we get the equation P(t) = 6.53(1.0114)t−2006 where t is years since 2006. (a) According to this model when will the population reach 9.25 billion people? (b) According to this model when will the population reach 11.75 billion people?

Section 6.7: Compound Interest

Summary We will be skipping this section.

Section 6.8: Exponential Growth and Decay; Newton’s Law; Etc

Summary We will applications of growth and decay with exponential equations; see how we can apply Newton’s Law; and look at logistic models.

Uninhibited Growth/Decay

Formula of Growth/Decay Many natural phenomena have been found to follow the law that an amount A varies with time t according to the function A(t) = A0ekt where A0 is the original (initial) amount when t = 0. If k > 0 then we have growth and if k < 0 we have decay.

Example A colony of bacteria grows according to the law of uninhibited growth according to the function N(t) = 90e0.05t where N is measured in grams and t is measure in days. (a) What is the initial amount of bacteria? (b) What is the growth rate of the bacteria? (c) What is the population after 5 days? (d) How long will it take for the population to reach 140 grams? (e) What is the doubling time for the population?

Example A colony of bacteria grows according to the law of uninhibited growth. (a) If the number of bacteria doubles in 4 hours, find the function that models the cells. (b) How long will it take for the colony to triple? (c) How long will it take to double in size again (increase four times)?

Example Traces of burned wood along with ancient stone tools in an archeological dig in Chile were found to contain approximately 1.67% of the original amount of carbon 14. If the half-life of carbon 14 is 5600 years, approximately when was the tree cut and burned?

Newton’s Law of Cooling

Newton’s Law of Cooling The temperature u of a heated object at a given time t can be modeled by the following function: u(t) = T + (u0 − T)ekt where T is the constant temperature of the surrounding medium, u0 is the

• riginal (initial) temperature, and k is a negative constant.

Example An object is heated to 100◦C and is then allowed to cool in a room whose air temperature is 30◦C. (a) If the temperature of an object is 80◦C after 5 minutes, when will its temperature be 50◦C? (b) How long will it take for the object to reach 35◦C.

Example A detective is called to the scene of a crime where a dead body has just been found. She arrives on the scene at 10:23 pm and begins her

• investigation. Immediately, the temperature of the body is taken and is

found to be 80◦F. The detective checks the programmable thermostat and finds that the room has been kept at a constant 68◦F for the past 3 days. After evidence from the crime scene is collected, the temperature of the body is taken once more and found to be 78.5◦F. This last temperature reading was taken exactly one hour after the first one. The next day the detective is asked by another investigator, “What time did our victim die?”

Logistic Model

Logistic Model In a logistic model, the population P after time t obeys the equation P(t) = c 1 + ae−bt where a, b, and c are constants with c > 0. The model is growth if b > 0 and decay if b < 0. The domain is all real numbers, and the range is (0, c). There are horizontal asymptotes of y = 0 and y = c; hence the range. We call c the carrying capacity as it is the maximum that a population can grow due to restraints.

Example Fruit flies are placed in a half-pin milk bottle with a banana (for food) and yeast (for a stimulus to lay eggs). Suppose that the fruit fly population after t days is given by P(t) = 230 1 + 56.6e−0.37t (a) State the carrying capacity and the growth rate. (b) Determine the initial population. (c) What is the population in 5 days? (d) How long will it take for the population to reach 180?

Example The logistic model P(t) = 0.9 1 + 6e−0.32t relate the proportion of U.S. households that own a DVD player to the

• year. Let t = 0 represent 200, t = 1 represent 2001 and so on.

(a) Determine the maximum proportion of households that will own a DVD player. (b) What proportion of households owned a DVD player in 2000? (c) What proportion of households owned a DVD player in 2010? (d) When will 85% of households own a DVD player?

Section 6.9: Building Exponential, Logarithmic, and Logistic Models from Data

Summary We will use data to generate equations to predict future

• utcomes. This is another form of regression. Recall that we

have done linear and quadratic regression.

Exponential Regression

Logarithmic Regression

Logistical Regression

Example Kathleen is interested in finding a function that explains the growth of cell phone usage in the United States. She gathers data on the number (in millions) of U.S. cell phone subscribers from 1985 through 2005. the data is shown on the next page. (a) Use the calculator to draw a scatter diagram. (b) Use the calculator to find an exponential function of best fit (regression equation). (c) Using this function predict the number of subscribers in 2012.

Example

Example