chapter exponential and logarithmic 12 . ... requires that the base be positive and different from...

CHAPTER

12 Exponential and Logarithmic Functions

12.1 Exponential Functions 12.2 Inverse Functions and Composite Functions 12.3 Logarithmic Functions 12.4 Properties of Logarithmic Functions 12.5 Natural Logarithmic Functions 12.6 Solving Exponential and Logarithmic Equations 12.7 Mathematical Modeling with Exponential and

Logarithmic Functions

OBJECTIVES

12.1 Exponential Functions

a Graph exponential equations and functions. b Graph exponential equations in which x and y have

been interchanged. c Solve applied problems involving applications of

exponential functions and their graphs.

Exponential Function

The function f(x) = ax, where a is a positive constant different from 1, is called an exponential function, base a.

EXAMPLE

a Graph exponential equations and functions.

1 Graph.

EXAMPLE Solution

Compute some function values and list the results in a table. It is a good idea to begin by letting x = 0.

EXAMPLE Solution

Next, plot these points and connect them with a smooth curve.

EXAMPLE Solution

As x increases, the function values increase indefinitely. As x decreases, the function values decrease, getting very close to 0. The x-axis, or the line y = 0, is an asymptote, meaning here that as x gets very small, the curve comes very close to but never touches the axis.

EXAMPLE

2 Graph.

EXAMPLE Solution

The definition of an exponential function, f(x) = ax, requires that the base be positive and different from 1.

When a > 1, the function f(x) = ax increases from left to right. The greater the value of a, the steeper the curve. As x gets smaller and smaller, the curve gets closer to the line y = 0: it is an asymptote.

y-Intercept of an Exponential Function

All functions f(x) = ax go through the point (0, 1). That is, the y-intercept is (0, 1).

EXAMPLE

3 Graph.

EXAMPLE Solution

Construct a table of values. Then plot the points and connect them with a smooth curve.

EXAMPLE Solution

EXAMPLE

b Graph exponential equations in which x and y have been interchanged.

5 Graph.

EXAMPLE Solution

c Solve applied problems involving applications of exponential functions and their graphs.

When interest is paid on interest, it is called compound interest.

In the second year, interest is earned on the first year’s interest as well as the original amount; the interest is compounded annually.

EXAMPLE

6 Interest Compounded Annually.

The amount of money A that a principal P will grow to after t years at interest rate r, compounded annually, is given by the formula

EXAMPLE

6 Interest Compounded Annually.

Suppose that $100,000 is invested at 4% interest, compounded annually. a) Find a function for the amount in the account after t

years. b) Find the amount of money in the account at t = 0, t = 4, t = 8, and t = 10. c) Graph the function.

EXAMPLE Solution

a) If P = $100,00 and r = 4% = 0.04, substitute these values and form the following function:

EXAMPLE Solution

b) To find the function values, you might find a calculator with a power key helpful.

EXAMPLE Solution

Compound-Interest Formula

If a principal P has been invested at interest rate r, compounded n times a year, in t years it will grow to an amount A given by

EXAMPLE

7 Compound Interest.

The Ibsens invest $4000 in an account paying compounded quarterly. Find the amount in the account after years.

EXAMPLE Solution

The amount in the account after years is $4270.39.

CHAPTER

OBJECTIVES

12.2 Inverse Functions and Composite Functions

a Find the inverse of a relation if it is described as a set of ordered pairs or as an equation.

b Given a function, determine whether it is one-to-one and has an inverse that is a function.

c Find a formula for the inverse of a function, if it exists, and graph inverse relations and functions.

d Find the composition of functions and express certain functions as a composition of functions.

OBJECTIVES

e Determine whether a function is an inverse by checking its composition with the original function.

A set of ordered pairs is called a relation.

Inverse Relation

Interchanging the coordinates of the ordered pairs in a relation produces the inverse relation.

EXAMPLE

1 Inverse Relation.

Consider the relation given by

In the figure, the relation is shown in red. The inverse of the relation is and is shown in blue.

EXAMPLE

1 Inverse Relation.

Inverse Relation

If a relation is defined by an equation, interchanging the variables produces an equation of the inverse relation.

EXAMPLE

3 Find an equation.

Find an equation of the inverse of the relation Then graph both the original relation and its inverse.

EXAMPLE Solution

One-To-One Function and Inverses

A function f is one-to-one if different inputs have different outputs— that is, if a ≠ b, then f(a) ≠ f(b). Or, A function f is one-to-one if when the outputs are the same, the inputs are the same—that is, if f(a) = f(b), then a = b.

One-To-One Function and Inverses

If a function is one-to-one, then its inverse is a function. The domain of a one-to-one function f is the range of the inverse f–1. The range of a one-to-one function f is the domain of the inverse f–1.

EXAMPLE

4 Graphs.

The graph of the exponential function f(x) = 2x, or y = 2x, is shown on the left. The graph of the inverse x = 2y is shown on the right.

EXAMPLE

4 Graphs.

EXAMPLE

4 Graphs.

The graph on the right passes the vertical-line test, so it is the graph of a function. However, look only at the graph on the left, think as follows:

EXAMPLE

4 Graphs.

A function is one-to-one if different inputs have different outputs. In other words, no two x-values will have the same y-value. For this function, we cannot find two x-values that have the same y-value. Note also that no horizontal line can be drawn that will cross the graph more than once. The function is thus one-to-one and its inverse is a function.

The Horizontal-Line Test

If it is possible for a horizontal line to intersect the graph of a function more than once, then the function is not one-to-one and therefore its inverse is not a function.

If a function f is one-to-one, a formula for its inverse f–1 can be found as follows: 1. Replace f(x) with y. 2. Interchange x and y. (This gives the inverse relation.) 3. Solve for y. 4. Replace y with f–1(x).

EXAMPLE

7 Find a formula and graph.

a) Determine whether the function is one-to-one. b) If it is one-to-one, find a formula for f–1(x). c) Graph the inverse function, if it exists.

Given:

EXAMPLE Solution

a) The graph passes the horizontal-line test and is one-to-one.

EXAMPLE Solution

The graph of f –1 is a reflection of the graph of f across the line y = x.

EXAMPLE

9 Find a formula and graph.

Consider. a) Determine whether the function is one-to-one. b) If it is one-to-one, find a formula for its inverse. c) Graph the inverse, if it exists.

EXAMPLE Solution

a) The graph of g(x) = x3 + 2 is shown in red. It passes the horizontal-line test and thus is one-to-one.

EXAMPLE Solution

c) To find the graph, reflect the graph of g(x) = x3 + 2 across the line y = x. It can also be found by substituting into and plotting points.

EXAMPLE Solution

Composite Function

EXAMPLE

10 Composite Functions.

EXAMPLE Solution

Check the values in part (a) with the formulas found in part (b):

EXAMPLE

12 Composite Functions.

EXAMPLE Solution

This is 7x + 3 to the 2nd power. Two functions that can be used for the composition are f(x) = x2 and g(x) = 7x + 3. Check by forming the composition:

If a function f is one-to-one, then f–1 is the unique function for which

EXAMPLE

13 Use composition.

EXAMPLE Solution

Find and check to see that each is x.

EXAMPLE Solution

CHAPTER

OBJECTIVES

12.3 Logarithmic Functions

a Graph logarithmic functions. b Convert from exponential equations to logarithmic

equations and from logarithmic equations to exponential equations.

c Solve logarithmic equations. d Find common logarithms on a calculator.

Meaning of Logarithms

Logarithms

a Graph logarithmic functions.

It is helpful in dealing with logarithmic functions to remember that the logarithm of a number is an exponent.

EXAMPLE

1 Graph.

EXAMPLE Solution

The equation y = log5x is equivalent to 5y = x. Find ordered pairs that are solutions by choosing values for y and computing the corresponding x-values.

EXAMPLE Solution

The table shows the following:

EXAMPLE Solution

Plot the ordered pairs and connect them with a smooth curve. The graph of y = 5x has been shown only for reference.

Converting Between Exponential and Logarithmic Equations.

EXAMPLE

b Convert from logarithmic equations to exponential equations.

2 Convert to a logarithmic equation.

EXAMPLE

Convert to logarithmic equations.

EXAMPLE Solution

EXAMPLE

5 Convert to an exponential equation.

EXAMPLE

Convert to an exponential equation.

EXAMPLE Solution

EXAMPLE

c Solve logarithmic equations.

8 Solve.

EXAMPLE Solution

Check: log2 is the exponent to which we raise 2 to get . Since 2–3 = , we know that checks and is the solution.

EXAMPLE

9 Solve.

EXAMPLE Solution

Check: log416 = 2 because 42 = 16. Thus, 4 is a solution. Since all logarithm bases must be positive, log–416 is not defined. Therefore, –4 is not a solution.

EXAMPLE

10 Find logarithm.

EXAMPLE Solution

Think of the meaning of log101000. It is the exponent to which we raise 10 to get 1000. That exponent is 3. Therefore, log101000 = 3.

Logarithm of 1

The Logarithm, Base a, of a

EXAMPLE

13 Simplify.

EXAMPLE Solution

d Find common logarithms on a calculator.

Base-10 logarithms are called common logarithms.

The abbreviation log, with no base written, is used for the common logarithm, base-10.

On a scientific or graphing calculator, the key for common logarithms is generally marked

EXAMPLE

Find the common logarithm.

Find the common logarithm, to four decimal places, on a scientific or graphing calculator.

EXAMPLE Solution

The logarithm of a negative number does not exist as a real number.

EXAMPLE

17 Complete the table.

Complete the following table to express each number in the first column as a power of 10. Round each exponent to the nearest ten-thousandth.

EXAMPLE

17 Complete the table.

EXAMPLE Solution

CHAPTER

OBJECTIVES

12.4 Properties of Logarithmic Functions

a Express the logarithm of a product as a sum of logarithms, and conversely.

b Express the logarithm of a power as a product. c Express the logarithm of a quotient as a difference of

logarithms, and conversely. d Convert from logarithms of products, quotients, and

powers to expressions in terms of individual logarithms, and conversely.

e Simplify expressions of the type logaak.

Property 1: The Product Rule

For any positive numbers M and N,

(The logarithm of a product is the sum of the logarithms of the factors. The number a can be any logarithm base.)

EXAMPLE

1 Express as a sum of logarithms.

EXAMPLE Solution

EXAMPLE

2 Express as a single logarithm.

EXAMPLE Solution

Property 2: The Power Rule

For any positive number M and any real number k, (The logarithm of a power of M is the exponent times the logarithm of M. The number a can be any logarithm base.)

EXAMPLE

b Express the logarithm of a power as a product.

Express as a product.

EXAMPLE Solution

b Express the logarithm of a power as a product.

Property 3: The Quotient Rule

For any positive numbers M and N, (The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator. The number a can be any logarithm base.)

EXAMPLE

c Express the logarithm of a quotient as a difference of logarithms, and conversely.

5 Express as a difference of logarithms.

EXAMPLE Solution

EXAMPLE

6 Express as a single logarithm.

EXAMPLE Solution

EXAMPLE

d Convert to expressions in terms of individual logarithms, and conversely.

Express.

EXAMPLE Solution

EXAMPLE

Express as a single logarithm.

EXAMPLE Solution

EXAMPLE

d Convert from logarithms of products, quotients, and powers.

EXAMPLE Solution

d Convert from logarithms of products, quotients, and powers..

EXAMPLE Solution

d Convert from logarithms of products, quotients, and powers..

For any base a, (The logarithm, base a, of a to a power is the power.)

Property 4

EXAMPLE

Simplify.

EXAMPLE Solution

CHAPTER

OBJECTIVES

12.5 Natural Logarithmic Functions

a Find logarithms or powers, base e, using a calculator. b Use the change-of-base formula to find logarithms

with bases other than e or 10. c Graph exponential and logarithmic functions, base e.

The Number e

On a calculator, the key for natural logarithms is generally marked

a Find logarithms or powers, base e, using a calculator.

Logarithms, base e, are called natural logarithms.

The abbreviation ln is commonly used with natural logarithms.

EXAMPLE

Find the natural logarithm.

Find the natural logarithm, to four decimal places, on a calculator.

EXAMPLE Solution

The inverse of a logarithmic function is an exponential function. Thus, if f(x) = lnx, then f–1(x) = ex. Because of this, on many calculators, the key doubles as the key after a or key has been pressed.

EXAMPLE

7 Find.

EXAMPLE Solution

The Change-of-Base Formula

EXAMPLE

b Use the change-of-base formula to find logarithms with bases other than e or 10.

8 Find.

EXAMPLE Solution

Let a = 10, b = 4, and M = 7. Then substitute into the change-of-base formula:

EXAMPLE

9 Find.

EXAMPLE Solution

EXAMPLE

c Graph exponential and logarithmic functions, base e.

11 Graph.

EXAMPLE Solution

EXAMPLE

14 Graph.

EXAMPLE Solution

CHAPTER

OBJECTIVES

12.6 Solving Exponential and Logarithmic Equations

a Solve exponential equations. b Solve logarithmic equations.

a Solve exponential equations.

Equations with variables in exponents are called exponential equations.

The Principle of Exponential Equality

(When powers are equal, the exponents are equal.)

For any

EXAMPLE

1 Solve.

EXAMPLE Solution

Note that 16 = 24. Thus write each side as a power of the same number:

Since the base is the same, 2, the exponents must be the same. Thus,

EXAMPLE Solution

The solution is 3.

The Principle of Logarithmic Equality

For any logarithm base a, and for x, y > 0, (If the logarithms, base a, of two expressions are the same, then the expressions are the same.)

EXAMPLE

2 Solve.

EXAMPLE Solution

EXAMPLE

3 Solve.

EXAMPLE Solution

b Solve logarithmic equations.

Equations containing logarithmic expressions are called logarithmic equations.

EXAMPLE

4 Solve.

To solve a logarithmic equation, first try to obtain a single logarithmic expression on one side and then write an equivalent exponential equation.

EXAMPLE

7 Solve.

EXAMPLE Solution

CHAPTER

OBJECTIVES

12.7 Mathematical Modeling with Exponential and Logarithmic Functions

a Solve applied problems involving logarithmic functions. b Solve applied problems involving exponential functions.

EXAMPLE

a Solve applied problems involving logarithmic functions.

1 Sound Levels.

To measure the “loudness” of any particular sound, the decibel scale is used. The loudness L, in decibels (dB), of a sound is given by where I is the intensity of the sound, in watts per square meter (W/m2), and ( is approximately the intensity of the softest sound that can be heard.)

EXAMPLE

1 Sound Levels.

a) An iPod can produce sounds of more than 10–0.5

W/m2, a volume that can damage the hearing of a person exposed to the sound for more than 28 sec. How loud, in decibels, is this sound level?

b) Audiologists and physicians recommend that earplugs be worn when one is exposed to sounds in excess of 90 dB. What is the intensity of such sounds?

EXAMPLE Solution

a) To find the loudness, in decibels, use the above formula:

The sound level is 115 decibels.

EXAMPLE Solution

b) Substitute and solve for I :

EXAMPLE Solution

Earplugs are recommended for sounds with intensities that exceed 10–3 W/m2.

EXAMPLE

2 Chemistry: pH of liquids.

In chemistry, the pH of a liquid is defined as pH = –log[H+], where [H+] is the hydrogen ion concentration in moles per liter.

a) The hydrogen ion concentration of human blood is normally about 3.98 10–8 moles per liter. Find the pH.

b) The pH of seawater is about 8.3. Find the hydrogen ion concentration.

EXAMPLE Solution

The pH of human blood is normally about 7.4.

a) To find the pH of human blood, use the above formula:

EXAMPLE Solution

b) Substitute and solve for [H+]:

The hydrogen ion concentration of seawater is about moles per liter.

Exponential Growth Model

An exponential growth model is a function of the form where P0 is the population at time 0, P(t) is the population at time t, and k is the exponential growth rate for the situation. The doubling time is the amount of time necessary for the population to double in size.

Exponential Growth Model

EXAMPLE

b Solve applied problems involving exponential functions.

4 Population Growth in India.

In 2009, India’s population was 1.166 billion, and the exponential growth rate was 1.55% per year.

a) Find the exponential growth function. b) What will the population be in 2015?

EXAMPLE Solution

a) We are trying to find a model. The given information allows us to create one. At t(0) (2009), the population was 1.166 billion. Substitute 1.166 for P0 and 1.55%, or

0.0155, for k to obtain the exponential growth function:

EXAMPLE Solution

b) In 2015, we have t = 6. That is, 6 yr have passed since 2009. To find the population in 2015, substitute 6 for t:

The population of India will be about 1.280 billion in

EXAMPLE

5 Interest Compounded Continuously

Suppose that an amount of money P0 is invested in a savings account at interest rate k, compounded continuously. That is, suppose that interest is computed every “instant” and added to the amount in the account. The balance P(t), after t years, is given by the exponential growth model:

EXAMPLE

5 Interest Compounded Continuously

a) Suppose that $30,000 is invested and grows to $34,855.03 in 5 years. Find the interest rate and then the exponential growth function.

b) What is the balance after 10 years? c) What is the doubling time?

EXAMPLE Solution

a) P0 = 30,000. Thus the exponential growth function is P(t) = 30,000ekt, where k must still be determined.

P(5) = 34,855.03. Substitute and solve for k:

EXAMPLE Solution

The interest rate is about 0.03, or 3%, compounded continuously. The exponential growth function is

EXAMPLE Solution

b) Substitute 10 for t: The balance in the account after 10 years will be

$40,495.76.

EXAMPLE Solution

c) To find the doubling time T, we replace P(t) with 60,000 and solve for T:

Thus the original investment of $30,000 will double in about 23.1 years.

EXAMPLE Solution

Exponential Decay Model

An exponential decay model is a function of the form P(t) = P0e–kt, k > 0, where P0 is the quantity present at time 0, P(t) is the amount present at time t, and k is the decay rate. The half-life is the amount of time necessary for half of the quantity to decay.

Exponential Decay Model

EXAMPLE

7 Carbon Dating.

The radioactive element carbon-14 has a half-life of 5750 yr. The percentage of carbon-14 present in the remains of organic matter can be used to determine the age of that organic matter. In a cave in Spain, archaeologists have found charcoal samples that have lost 96.5% of their carbon-14. The age of these samples suggests that Neanderthals were in existence about 4000 yr longer than had been previously thought. What is the age of the samples?

EXAMPLE Solution

First find k. To do so, use the concept of half-life. When t = 5750 (the half-life), P(t) will be half of P0. Then

EXAMPLE Solution

The function for the decay of carbon-14:

If the charcoal has lost 96.5% of its carbon-14 from an initial amount P0, then 100% – 96.5%, or 3.5% of P0 is still present. To find the age t of the charcoal, solve the following equation for t:

EXAMPLE Solution

The charcoal samples are about 28,000 yr old.

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