reply to each of the discussion post. each post response should be a minimum of ONE or TWO each.

Ch4lecture.pptx

Home >Mathematics homework help >Algebra homework help >reply to each of the discussion post. each post response should be a minimum of ONE or TWO each.

‹#›

Inverse Functions

SECTION 4.1

Click to edit Master text styles

Second level

Third level

Fourth level

Fifth level

‹#›

Introduction to Inverse Functions

‹#›

Introduction to Inverse Functions

Consider the “doubling function” f (x) = 2x that doubles every input. Some of the ordered pairs of this function are

(–4, –8), (–1.5, –3), (1, 2), , (7, 14)

Now consider the “halving function” g(x) = that takes

one-half of every input. Some of the ordered pairs of this

function are

(–8, –4), (–3, –1.5), (2, 1), , (14, 7)

‹#›

Introduction to Inverse Functions

Observe that the ordered pairs of g are the ordered pairs of f with the order of the coordinates reversed. The following two examples illustrate this concept.

f (5) = 2(5) = 10 g(10) = (10) = 5

Ordered pair: (5, 10) Ordered pair: (10, 5)

f (a) = 2(a) = 2a g(2a) = (2a) = a

Ordered pair: (a, 2a) Ordered pair: (2a, a)

The function g is said to be the inverse function of f.

‹#›

Introduction to Inverse Functions

Definition of an Inverse Function

If the ordered pairs of a function g are the ordered pairs of a function f with the order of the coordinates reversed, then g is the inverse function of f.

Consider a function f and its inverse function g. Because the ordered pairs of g are the ordered pairs of f with the order of the coordinates reversed, the domain of the inverse function g is the range of f, and the range of g is the domain of f.

‹#›

Introduction to Inverse Functions

Not all functions have an inverse that is a function. Consider, for instance, the “square function” S(x) = x2. Some of the ordered pairs of S are

{(–3, 9), (–1, 1), (0, 0), (1, 1), (3, 9), (5, 25)}

If we reverse the coordinates of the ordered pairs, we have

{(9, –3), (1, –1), (0, 0), (1, 1), (9, 3), (25, 5)}

‹#›

Introduction to Inverse Functions

This set of ordered pairs is not a function because there are ordered pairs, for instance (9, –3) and (9, 3), with the same first coordinate and different second coordinates.

In this case, S has an inverse relation but not an inverse function.

A graph of S is shown in Figure 4.1.

Figure 4.1

‹#›

Introduction to Inverse Functions

Note that x = –3 and x = 3 produce the same value of y.

Thus the graph of S fails the horizontal line test; therefore, S is not a one-to-one function.

This observation is used in the following theorem.

Condition for an Inverse Function

A function f has an inverse function if and only if f is a one-to-one function.

‹#›

Introduction to Inverse Functions

We know that increasing functions and decreasing functions are one-to-one functions. Thus we can state the following theorem.

A Property of Increasing Functions and Decreasing Functions

If f is an increasing function or a decreasing function, then f has an inverse function.

If a function g is the inverse of a function f, we usually denote the inverse function by f –1 rather than g. For the doubling and halving functions f and g, we write

f (x) = 2x f –1(x) =

‹#›

Graphs of Inverse Functions

‹#›

Graphs of Inverse Functions

Because the coordinates of the ordered pairs of the inverse of a function f are the ordered pairs of f with the order of the coordinates reversed, we can use them to create a graph of f –1.

‹#›

Example 1 – Sketch the Graph of the Inverse of a Function

Sketch the graph of f –1 given that f is the function shown in Figure 4.2.

Figure 4.2

‹#›

Example 1 – Solution

Because the graph of f passes through (–1, 0.5), (0, 1), (1, 2), and (2, 4), the graph of f –1 must pass through (0.5, –1), (1, 0), (2, 1), and (4, 2).

Plot the points and then draw a smooth curve through the points, as shown in Figure 4.3.

Figure 4.3

‹#›

Graphs of Inverse Functions

The graph from the solution to Example 1 is shown again in Figure 4.4.

Note that the graph of f –1 is symmetric to the graph of f with respect to the graph of y = x.

If the graph were folded along the dashed line, the graph of f would lie on top of the graph of f –1.

Figure 4.4

‹#›

Graphs of Inverse Functions

This is a characteristic of all graphs of functions and their inverses.

In Figure 4.5, although S does not have an inverse that is a function, the graph of the inverse relation S –1 is symmetric to S with respect to the graph of y = x.

Figure 4.5

‹#›

Composition of a Function and Its Inverse

‹#›

Composition of a Function and Its Inverse

Observe the effect of forming the composition of f (x) = 2x

and g(x) =

f (x) = 2x g(x) =

f [g(x)] = g[f (x)] =

f [g(x)] = x g[f (x)] = x

Replace x

with g(x).

Replace x

with f (x).

‹#›

Composition of a Function and Its Inverse

This property of the composition of inverse functions always holds true.

When taking the composition of inverse functions, the inverse function reverses the effect of the original function.

For the two functions above, f doubles a number, and g halves a number.

If you double a number and then take one-half of the result, you are back to the original number.

‹#›

Composition of a Function and Its Inverse

Composition of Inverse Functions Property

If f is a one-to-one function, then f –1 is the inverse function of f if and only if

(f  f –1)(x) = f [f –1(x)] = x for all x in the domain of f –1

and

(f –1  f )(x) = f –1[f (x)] = x for all x in the domain of f.

‹#›

Example 2 – Use the Composition of Inverse Functions Property

Use composition of functions to show that f –1(x) = 3x – 6 is

the inverse function of

Solution:

We must show that f [f –1(x)] = x and f –1[f (x)] = x.

f –1(x) = 3x – 6

f [f –1(x)] = x f –1[f (x)] = x

‹#›

Finding an Inverse Function

‹#›

Finding an Inverse Function

If a one-to-one function f is defined by an equation, then we can use the following method to find the equation for f –1.

Steps for Finding the Inverse of a Function

To find the equation of the inverse f –1 of the one-to-one function f, follow these steps.

1. Substitute y for f (x).

2. Interchange x and y.

3. Solve, if possible, for y in terms of x.

4. Substitute f –1(x) for y.

‹#›

Example 3 – Find the Inverse of a Function

Find the inverse of f (x) = 3x + 8.

Solution:

f (x) = 3x + 8

y = 3x + 8

x = 3y + 8

x – 8 = 3y

Replace f (x) with y.

Interchange x and y.

Solve for y.

‹#›

Example 3 – Solution

The inverse function is given by

Replace y with f –1.

cont’d

‹#›

Finding an Inverse Function

The graph of f (x) = x2 + 4x + 3 is shown in Figure 4.9a.

The function f is not a one-to-one function and therefore does not have an inverse function.

Figure 4.9(a)

‹#›

Finding an Inverse Function

However, the function given by G(x) = x2 + 4x + 3, shown in Figure 4.9b, for which the domain is restricted to {x | x  –2}, is a one-to-one function and has an inverse function G –1.

Figure 4.9(b)

‹#›

Exponential Functions and Their Applications

SECTION 4.2

Click to edit Master text styles

Second level

Third level

Fourth level

Fifth level

‹#›

Exponential Functions

‹#›

Exponential Functions

When a parking facility opened in 1980, it charged $1 for all-day parking.

Since then, it has doubled its daily parking fee every 8 years as shown in Table 4.1.

Table 4.1

‹#›

Exponential Functions

In Figure 4.13, we have plotted the data in Table 4.1 and modelled the upward trend in the parking fee by a smooth curve.

This model is based on an exponential function.

The effectiveness of a drug that is used for sedation during a surgical procedure depends on the concentration of the drug in the patient.

Figure 4.13

‹#›

Exponential Functions

Through natural body chemistry, the amount of this drug in the body decreases over time.

The graph in Figure 4.14 models this decrease. This model is another example of an exponential model.

Figure 4.14

‹#›

Exponential Functions

Definition of an Exponential Function

The exponential function with base b is defined by

f (x) = bx

where b > 0, b  1, and x is a real number.

The base b of f (x) = bx is required to be positive.

If the base were a negative number, the value of the function would be a complex number for some values of x.

‹#›

Exponential Functions

For instance, if b = – 4 and , then

To avoid complex number values of a function, the base of any exponential function must be a positive number.

Also, b is defined such that b  1 because f (x) = 1x = 1 is a constant function.

You may have noticed that in the definition of an exponential function the exponent x is a real number.

‹#›

Exponential Functions

We have already worked with expressions of the form bx, where b > 0 and x is a rational number. For instance,

23 = 2  2  2 = 8

272/3 = ( )2 = 32 = 9

320.4 = 322/5 = ( )2 = 22 = 4

‹#›

Example 1 – Evaluate an Exponential Function

Evaluate f (x) = 3x at x = 2, x = –4, and x = .

Solution:

f (2) = 32

f (–4) = 3–4

f () = 3

 33.1415927

 31.54428

= 9

Evaluate with the aid of a calculator.

‹#›

Graphs of Exponential Functions

‹#›

Graphs of Exponential Functions

The graph of f (x) = 2x is shown in Figure 4.15. The coordinates of some of the points on the curve are given in Table 4.2.

Figure 4.15

Table 4.2

‹#›

Graphs of Exponential Functions

Note the following properties of the graph of the exponential function f (x) = 2x.

The y–intercept is (0, 1).

The graph passes through (1, 2).

As x decreases without bound (that is, as x ), f (x) 0.

The graph is a smooth, continuous increasing curve.

Now consider the graph of an exponential function for

which the base is between 0 and 1.

‹#›

Graphs of Exponential Functions

The graph of is shown in Figure 4.16. The

coordinates of some of the points on the curve are given in

Table 4.3.

Figure 4.16

Table 4.3

‹#›

Graphs of Exponential Functions

Note the following properties of the graph of .

The y–intercept is (0, 1).

The graph passes through

As x increases without bound (that is, as x ), f (x) 0.

The graph is a smooth, continuous decreasing curve.

‹#›

Graphs of Exponential Functions

The basic properties of exponential functions are provided in the following summary.

Properties of f (x) = bx

For positive real numbers b, b  1, the exponential function defined by f (x) = bx has the following properties:

The function f is a one–to–one function. It has the set of real numbers as its domain and the set of positive real numbers as its range.

The graph of f is a smooth, continuous curve with a y–intercept of (0, 1), and the graph passes through (1, b).

‹#›

Graphs of Exponential Functions

If b > 1, f is an increasing function and the graph of f is asymptotic to the negative x–axis. [As and as ] See Figure 4.17a.

Figure 4.17(a)

f (x) = bx, b > 1

‹#›

Graphs of Exponential Functions

If 0 < b < 1, f is a decreasing function and the graph of f is asymptotic to the positive x–axis. [As and as ] See Figure 4.17b.

Figure 4.17(b)

f (x) = bx, 0 < b < 1

‹#›

Example 2 – Graph an Exponential Function

Graph: g(x) =

Solution:

Because the base is less than 1, we know that the graph

of g is a decreasing function that is asymptotic to the

positive x–axis.

The y–intercept of the graph is the point (0, 1), and the

graph passes through

‹#›

Example 2 – Solution

Plot a few additional points (see Table 4.4), and then draw a smooth curve through the points, as in Figure 4.18.

cont’d

Table 4.4

Figure 4.18

‹#›

Graphs of Exponential Functions

The graphs of some functions can be constructed by stretching, compressing, or reflecting the graph of an exponential function.

‹#›

Natural Exponential Function

‹#›

Natural Exponential Function

The irrational number  is often used in applications that involve circles. Another irrational number, denoted by the letter e, is useful in many applications that involve growth or decay.

Definition of e

The letter e represents the number that

approaches as n increases without bound.

The letter e was chosen in honor of the Swiss mathematician Leonhard Euler.

‹#›

Natural Exponential Function

He was able to compute the value of e to several decimal

places by evaluating for large values of n, as

shown in Table 4.5.

The value of e accurate to eight decimal places is

2.71828183.

Table 4.5

‹#›

Natural Exponential Function

The base of an exponential function can be any positive real number other than 1.

The number 10 is a convenient base to use for some situations, but we will see that the number e is often the best base to use in real–life applications.

The exponential function with e as the base is known as the natural exponential function.

‹#›

Natural Exponential Function

Definition of the Natural Exponential Function

For all real numbers x, the function defined by

f (x) = ex

is called the natural exponential function.

A calculator can be used to evaluate ex for specific values of x. For instance,

e2  7.389056, e3.5  33.115452, and e–1.4  0.246597

On a TI-83/TI-83 Plus/TI-84 Plus calculator, the ex function is located above the key.

‹#›

Natural Exponential Function

To graph f (x) = ex, use a calculator to find the range values for a few domain values.

The range values in Table 4.6 have been rounded to the nearest tenth.

Table 4.6

‹#›

Natural Exponential Function

Plot the points given in Table 4.6, and then connect the points with a smooth curve. Because e > 1, we know that the graph is an increasing function.

To the far left, the graph will approach the x–axis.

The y–intercept is (0, 1). See Figure 4.23.

Figure 4.23

‹#›

Natural Exponential Function

Note in Figure 4.24 how the graph of f (x) = ex compares with the graphs of g(x) = 2x and h(x) = 3x.

You may have anticipated that the graph of f (x) = ex would lie between the two other graphs because e is between 2 and 3.

Figure 4.24

‹#›

Logarithmic Functions and Their Applications

SECTION 4.3

Click to edit Master text styles

Second level

Third level

Fourth level

Fifth level

‹#›

Logarithmic Functions

‹#›

Logarithmic Functions

Every exponential function of the form g(x) = bx is a one-to-one function and therefore has an inverse function.

Sometimes we can determine the inverse of a function represented by an equation by interchanging the variables of its equation and then solving for the dependent variable.

If we attempt to use this procedure for g(x) = bx, we obtain

g(x) = bx

y = bx

x = by

Interchange the variables.

‹#›

Logarithmic Functions

One method would be to merely write

y = the power of b that produces x

Although this would work, it is not concise. We need a compact notation to represent “y is the power of b that produces x.” This more compact notation is given in the following definition.

‹#›

Logarithmic Functions

Definition of a Logarithm and a Logarithmic Function

If x > 0 and b is a positive constant (b ≠ 1), then

y = logb x if and only if by = x

The notation logb x is read “the logarithm (or log) base b of x.”

The function defined by f(x) = logb x is a logarithmic function with base b. This function is the inverse of the exponential function g(x) = bx.

‹#›

Logarithmic Functions

It is essential to remember that f (x) = logb x is the inverse function of g(x) = bx.

Because these functions are inverses and because functions that are inverses have the property that f (g(x)) = x and g(f (x)) = x, we have the following important relationships.

Composition of Logarithmic and Exponential Functions

Let g(x) = bx and f (x) = logb x (x > 0, b > 0, b ≠ 1). Then

and

‹#›

Logarithmic Functions

As an example of these relationships, let g(x) = 2x and f (x) = log2 x. Then

and

The equations

y = logb x and by = x

are different ways of expressing the same concept.

‹#›

Logarithmic Functions

Definition of Exponential Form and Logarithmic Form

The exponential form of y = logb x is by = x.

The logarithmic form of by = x is y = logb x.

‹#›

Example 1 – Change from Logarithmic to Exponential Form

Write each equation in its exponential form.

a. 3 = log2 8

b. 2 = log10(x + 5)

c. loge x = 4

d. logb b3 = 3

‹#›

Example 1 – Solution

Use the definition y = logb x if and only if by = x.

b. 2 = log10(x + 5) if and only if 102 = x + 5.

c. loge x = 4 if and only if e4 = x.

d. logb b3 = 3 if and only if b3 = b3.

‹#›

Logarithmic Functions

The definition of a logarithm and the definition of an inverse

function can be used to establish many properties of

logarithms. For instance,

logb b = 1 because b = b1.

logb 1 = 0 because 1 = b0.

logb(bx) = x because bx = bx.

because f (x) = logb x and g(x) = bx are inverse functions. Thus g[f (x)] = x.

‹#›

Logarithmic Functions

We will refer to the preceding properties as the basic logarithmic properties.

Basic Logarithmic Properties

1. logb b = 1

2. logb 1 = 0

3. logb(bx) = x

‹#›

Logarithmic Functions

Some logarithms can be evaluated just by remembering that a logarithm is an exponent.

For instance, log5 25 equals 2 because the base 5 raised to the second power equals 25.

log10 100 = 2 because 102 = 100.

log4 64 = 3 because 43 = 64.

because

‹#›

Graphs of Logarithmic Functions

‹#›

Graphs of Logarithmic Functions

Because f(x) = logb x is the inverse function of g(x) = bx, the graph of f is a reflection of the graph of g across the line given by y = x.

The graph of g(x) = 2x is shown in Figure 4.25.

Figure 4.25

‹#›

Graphs of Logarithmic Functions

Table 4.7 shows some of the ordered pairs of the graph of g.

The graph of the inverse of g, which is f (x) = log2 x, is also shown in Figure 4.25. Some of the ordered pairs of f are shown in Table 4.8.

Table 4.7

Table 4.8

‹#›

Graphs of Logarithmic Functions

Note that if (x, y) is a point on the graph of g, then (y, x) is a

point on the graph of f. Also notice that the graph of f is a

reflection of the graph of g across the line given by y = x.

The graph of a logarithmic function can be drawn by first

rewriting the function in its exponential form. This

procedure is illustrated in Example 4.

‹#›

Example 4 – Graph a Logarithmic Function

Graph f (x) = log3 x.

Solution:

To graph f (x) = log3 x, consider the equivalent exponential equation x = 3y.

Because this equation is solved for x, choose values of y and calculate the corresponding values of x, as shown in Table 4.9.

Table 4.9

‹#›

Example 4 – Solution

Now plot the ordered pairs and connect the points with a smooth curve, as shown in Figure 4.26.

Figure 4.26

cont’d

‹#›

Graphs of Logarithmic Functions

We can use a similar procedure to draw the graph of a logarithmic function with a fractional base.

For instance, consider y = log2/3 x. Rewriting this in

exponential form gives us

Choose values of y and calculate the corresponding x values. See Table 4.10.

Table 4.10

‹#›

Graphs of Logarithmic Functions

Plot the points corresponding to the ordered pairs (x, y), and then draw a smooth curve through the points, as shown in Figure 4.27.

Figure 4.27

‹#›

Graphs of Logarithmic Functions

Properties of f (x) = logb x

For all positive real numbers b, b ≠ 1, the function f (x) = logb x has the following properties.

The domain of f consists of the set of positive real numbers, and its range consists of the set of all real numbers.

The graph of f has an x-intercept of (1, 0) and passes through (b, 1).

‹#›

Graphs of Logarithmic Functions

If b > 1, f is an increasing function and its graph is asymptotic to the negative y-axis. [As x  , f (x)  , and as x  0 from the right, f (x)  – .] See Figure 4.28a.

Figure 4.28(a)

f (x) = logb x, b > 1

‹#›

Graphs of Logarithmic Functions

If 0 < b < 1, f is an decreasing function and its graph is asymptotic to the positive y-axis. [As x  , f (x)  – , and as x  0 from the right, f(x)  .] See Figure 4.28b.

Figure 4.28(b)

f (x) = logb x, 0 < b < 1

‹#›

Domains of Logarithmic Functions

‹#›

Domains of Logarithmic Functions

The function f (x) = logb x has as its domain the set of

positive real numbers.

The function f (x) = logb (g(x)) has as its domain the set of all x for which g(x) > 0.

To determine the domain of a function such as f (x) = logb (g(x)), we must determine the values of x that make g(x) positive. This process is illustrated in Example 5.

‹#›

Example 5 – Find the Domain of a Logarithmic Function

Find the domain of each of the following logarithmic functions.

a. f (x) = log6(x – 3)

b. F(x) = log2 |x + 2|

‹#›

Example 5 – Solution

a. Solving (x – 3) > 0 for x gives us x > 3.

The domain of f consists of all real numbers greater than 3.

In interval notation, the domain is

b. The solution set of | x + 2 | > 0 consists of all real numbers x except x = –2.

The domain of F consists of all real numbers x ≠ –2.

In interval notation, the domain is

‹#›

Example 5 – Solution

c. Solving yields the set of all real numbers x

between 0 and 8.

The domain of R is all real numbers x such that 0 < x < 8.

In interval notation, the domain is (0, 8).

cont’d

‹#›

Common and Natural Logarithms

‹#›

Common and Natural Logarithms

Two of the most frequently used logarithmic functions are common logarithms, which have base 10, and natural logarithms, which have base e (the base of the natural exponential function).

Definition of Common and Natural Logarithms

The function defined by f (x) = log10 x is called the common logarithmic function.

It is customarily written as f (x) = log x, without stating the base.

‹#›

Common and Natural Logarithms

The function defined by f (x) = loge x is called the natural logarithmic function.

It is customarily written as f (x) = ln x.

Most scientific or graphing calculators have a key for evaluating common logarithms and an key to evaluate natural logarithms.

For instance, using a graphing calculator,

log 24  1.3802112 and ln 81  4.3944492

‹#›

Properties of Logarithms and Logarithmic Scales

SECTION 4.4

Click to edit Master text styles

Second level

Third level

Fourth level

Fifth level

‹#›

Properties of Logarithms

‹#›

Properties of Logarithms

We have introduced the following basic properties of logarithms.

logb b = 1 and logb 1 = 0

Also, because exponential functions and logarithmic functions are inverses of each other, we observed the relationships

logb(bx) = x and blogb x = x

‹#›

Properties of Logarithms

We can use the properties of exponents to establish the following additional logarithmic properties.

Properties of Logarithms In the following properties, b, M, and N are positive real numbers (b  1).

Product property logb(MN) = logb M + logb N

Quotient property logb = logb M – logb N

Power property logb(M p) = p logb M

‹#›

Properties of Logarithms

Logarithm-of-each-side property M = N implies logb M = logb N

One-to-one property logb M = logb N implies M = N

The properties of logarithms are often used to rewrite logarithmic expressions in an equivalent form.

‹#›

Properties of Logarithms

The process of using the product or quotient properties to rewrite a single logarithm as the sum or difference of two or more logarithms, or using the power property to rewrite logb(M p) in its equivalent form p logb M, is called expanding the logarithmic expression.

We illustrate this process in Example 1.

‹#›

Example 1 – Expand Logarithmic Expressions

Use the properties of logarithms to expand the following logarithmic expressions. Assume all variable expressions represent positive real numbers. When possible, evaluate logarithmic expressions.

a. log5(xy2) b.

Solution:

a. log5(xy2) = log5 x + log5 y2

= log5 x + 2 log5 y

Product property

Power property

‹#›

Example 1 – Solution

cont’d

Quotient property

Product property

Power property

Evaluate ln e.

Write as .

‹#›

Properties of Logarithms

The properties of logarithms are also used to condense expressions that involve the sum or difference of logarithms into a single logarithm.

For instance, we can use the product property to rewrite

logb M + logb N as logb(MN), and the quotient property to

rewrite logb M – logb N as logb .

Before applying the product or quotient properties, use the power property to write all expressions of the form p logb M in their equivalent logb M p form.

‹#›

Change-of-Base Formula

‹#›

Change-of-Base Formula

Recall that to determine the value of y in log3 81 = y, we ask the question, “What power of 3 is equal to 81?” Because 34 = 81, we have log3 81 = 4.

Now suppose that we need to determine the value of log3 50. In this case, we need to find the power of 3 that produces 50.

Because 33 = 27 and 34 = 81, the value we are seeking is somewhere between 3 and 4.

‹#›

Change-of-Base Formula

The following procedure can be used to produce an estimate of log3 50.

The exponential form of log3 50 = y is 3y = 50. Applying logarithmic properties gives us

3y = 50

ln 3y = ln 50

y ln 3 = ln 50

y =



Logarithm-of-each-side property

Power property

Solve for y.

‹#›

Change-of-Base Formula

Thus log3 50  3.56088. In the preceding procedure we could just as well have used logarithms of any base and arrived at the same value.

Thus any logarithm can be expressed in terms of logarithms of any base we wish. This general result is summarized in the following formula.

Change-of-Base Formula

If x, a, and b are positive real numbers with a  1 and b  1, then

‹#›

100

Change-of-Base Formula

Because most calculators use only common logarithms (a = 10) or natural logarithms (a = e), the change-of-base formula is used most often in the following form.

If x and b are positive real numbers and b  1, then

‹#›

101

Example 3 – Use the Change-of-Base Formula

Evaluate each logarithm. Round to the nearest ten- thousandth.

a. log3 18 b. log12 400

Solution:

To approximate these logarithms, we may use the change-of-base formula with a = 10 or a = e. For this example, we choose to use the change-of-base formula with a = e.

That is, we will evaluate these logarithms by using the key on a scientific or graphing calculator.

‹#›

102

Example 3 – Solution

a. log3 18 =

 2.6309

b. log12 400 =

 2.4111

cont’d

‹#›

103

Change-of-Base Formula

The change-of-base formula and a graphing calculator can be used to graph logarithmic functions that have a base other than 10 or e.

For instance, to graph f (x) = log3(2x + 3), we rewrite the function in terms of base 10 or base e.

Using base 10 logarithms, we have

f (x) = log3(2x + 3)

The graph is shown in Figure 4.33.

f (x) = log3(2x + 3)

Figure 4.33

‹#›

104

Exponential and Logarithmic Equations

SECTION 4.5

Click to edit Master text styles

Second level

Third level

Fourth level

Fifth level

‹#›

105

Solving Exponential Equations

‹#›

106

Solving Exponential Equations

If a variable appears in the exponent of a term of an equation, such as in 2x + 1 = 32, then the equation is called an exponential equation.

Example 1 uses the following Equality of Exponents Theorem to solve 2x + 1 = 32.

Equality of Exponents Theorem

If bx = by, then x = y, provided b > 0 and b  1.

‹#›

107

Example 1 – Solve an Exponential Equation

Use the Equality of Exponents Theorem to solve 2x + 1 = 32.

Solution:

2x + 1 = 32

2x + 1 = 25

x + 1 = 5

x = 4

Check: Let x = 4. Then 2x + 1 = 24 + 1

= 25

= 32

Write each side as a power of 2.

Equate the exponents.

Solve for x.

‹#›

108

Solving Logarithmic Equations

‹#›

109

Solving Logarithmic Equations

Equations that involve logarithms are called logarithmic equations.

The properties of logarithms, along with the definition of a logarithm, are often used to find the solutions of a logarithmic equation.

‹#›

110

Example 5 – Solve a Logarithmic Equation

Solve: log(3x – 5) = 2

Solution:

log(3x – 5) = 2

3x – 5 = 102

3x = 105

x = 35

Check: log[3(35) – 5] = log 100

= 2

Definition of a logarithm

Solve for x.

‹#›

111