Exponential and Logarithmic Graphs...Lesson 24: Graphing Logarithmic Functions 229onnect Duplicating...

228 Unit 5: Exponential and Logarithmic Functions

LESSON

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UNDERSTAND The graph of the exponential function y 5 2x is shown. There is a horizontal asymptote at y 5 0 (or the x-axis).

Logarithmic functions are the inverses of exponential functions. Recall that the inverse of a function can be found by exchanging x and y and then solving for y.

y 5 2x Exchange x and y.

x 5 2y Write the related logarithmic equation.

y 5 log2 x

The graph of y 5 log2 x is shown. It has a vertical asymptote at x 5 0 (or the y-axis). Look at the graph’s end behavior. As x approaches 0 from the right, y approaches 2`; as x approaches `, y approaches `.

Now both graphs are shown on the same coordinate plane. The dashed line shows y 5 x. The graphs are reflections of each other over this line because they are inverse functions.

Exponential and Logarithmic Graphs

x

y

–2 2 4 6 8 100

–2

2

4

6

8

10

y � 2x

x

y

–2 2 4 6 8 100

–2

2

4

6

8

10

y � log2 x

x

y

–2 2 4 6 8 100

–2

2

4

6

8

10

y � 2x

y � x

y � log2 x

Graphing Logarithmic Functions24

Lesson 24: Graphing Logarithmic Functions 229

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The graph of y 5 3x 1 1 is shown. Find and graph its inverse function.

y

x–4 –2

–2

–4

2

2

4

6

8

10

12

14

4 6 8 10 12 140

Find the inverse function.

Switch x and y.

x 5 3y 1 1

Isolate the exponential expression.

x 2 1 5 3y

Write the related logarithmic function.

▸y 5 log3 (x 2 1)

Graph the inverse function.

Plot the ordered pairs and connect them.

▸ y

x–4 –2

–2

–4

2

2

4

6

8

10

12

14

4 6 8 10 12 140

3

1

Find ordered pairs for the inverse function.

Rather than finding values of log3 (x 2 1) for different values of x, find values for the exponential function. Then, switch the x- and y-values.

x 3x 1 1

21 4 __ 3

0 2

1 4

2 10

The graph of the inverse function will include the points ( 4 __ 3 , 21 ) , (2, 0), (4, 1), and (10, 2).

2

What are the asymptotes of each graph? What can you conclude about the asymptotes of a graph and its inverse?

DISCUSS


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UNDERSTAND One form of a general logarithmic equation is y 5 a log [b(x 2 h)] 1 k. The parameters involved, a, b, h, and k, have different effects on the function’s graph.

The parameter h produces a horizontal translation of the graph by h units. This translation is to the right for h . 0 and to the left for h , 0. This translation includes the vertical asymptote, so the asymptote of the function is x 5 h.

Since log 1 5 0, the x-intercept is the value where b(x 2 h) 5 1. So, the x-intercept is ( 1 __ b 1 h, 0 ) .

x

y

–2–4–6 2 4 60

–2

–4

2

4

6

y � log x

y � log (x � 4)

y � log (x � 2)

–6

UNDERSTAND The parameter b has different effects on the graph of a logarithmic equation in the form y 5 a log [b(x 2 h)] 1 k.

• If b . 0, the entire graph is to the right of the vertical asymptote.

• If b , 0, the entire graph is to the left of the vertical asymptote.

The parameter b horizontally stretches or shrinks the graph of a logarithmic equation from or

toward the vertical asymptote, x 5 h. Multiply all x-values by 1 __ b .

• If |b| , 1, the graph is stretched horizontally.

• If |b| . 1, the graph is shrunk horizontally.

x

y

–2–4–6 2 4 60

–2

–4

2

4

6

y � log xy � log �x

–6

y � log x3

Horizontal Transformations of Logarithmic Functions


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Graph f (x) 5 log2 4(x 2 3), and analyze key features of the graph.

Calculate the x-intercept.

Use ( 1 __ b 1 h, 0 ) with b 5 4 and h 5 3.

1 __ 4 1 3 5 3.25.

The x-intercept is (3.25, 0), as shown on the graph.

4

Find points using a table.

To evaluate, simplify the argument of the logarithm, and then use the change of base formula.

x log2 4(x 2 3) y

3.5 log2 2 1

4 log2 4 2

5 log2 8 3

7 log2 16 4

11 log2 32 5

Identify key features.

Notice that the asymptote is the vertical line x 5 3. This corresponds to the value of h in the function.

The value of b in the function is 4. So, the graph is shrunk horizontally toward the asymptote by 4.

3

1

Plot the points to sketch the graph.

x

y

2–2 4 6 8 10 120

–2

–4

2

4

6

2

Find the x-intercept and asymptote of f (x) 5 log5 2(x 1 4).

TRY


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UNDERSTAND You also can consider the unique effects that the parameters k and a in the general logarithmic equation y 5 a log [b(x 2 h)] 1 k have on the graph of a logarithmic function. The parameter k produces a vertical translation of the graph by k units. This translation is up for k . 0 and down for k , 0.

The parameter a stretches or shrinks the graph of a logarithmic equation from or toward the line y 5 k by a.

• If |a| , 1, the graph is shrunk vertically.

• If |a| . 1, the graph is stretched vertically.

Look at the graph of y 5 log x. You can rewrite that equation as y 5 a log [b(x 2 h)] 1 k, with a and b equal to 1 and h and k equal to 0. Now examine the effect of the parameter a on the graph. When a 5 1, as x approaches h (or 0, the vertical asymptote), values of y approach 2`. But when a 5 21, as shown in the graph of y 5 2log x, as x approaches h (or 0, the vertical asymptote), values of y approach `.

UNDERSTAND You can use the product property of logarithms to rewrite the general form logarithmic equation y 5 a log [b(x 2 h)] 1 k.

y 5 a log [b(x 2 h)] 1 k

y 5 a [log b 1 log (x 2 h)] 1 k

y 5 [a log b 1 a log (x 2 h)] 1 k

Now let j 5 a log b.

y 5 j 1 a log (x 2 h) 1 k or y 5 a log (x 2 h) 1 j 1 k

Using the form y 5 a log (x 2 h) 1 j 1 k, you can identify any vertical translation more easily. The graph of the function is moved (j 1 k) units up or down.

Vertical Transformations of Logarithmic Functions

x

y

–2–4–6 2 4 60

–2

–4

2

4

6

y � log x

y � 3 log x

y � �log x

–6

x

y

–2–4–6 2 4 60

–2

–4

2

4

6

y � log x

y � log x � 3

y � log x � 4

–6


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Graph f (x) 5 2log3 27x, and analyze key features of the graph.

Calculate the x-intercept.

Use ( 1 __ b 1 h, 0 ) with b 5 27 and h 5 0.

1 ___ 27 1 0 5 1

___ 27

The x-intercept is ( 1 ___ 27 , 0 ) . This can be

confirmed on the graph.

4

Create a table of ordered pairs for the function.

Evaluate the logarithm for values of x that make the argument, 27x, equal to a power of the base, 3.

x log3 27x y

1 __ 9 log3 3 21

1 __ 3 log3 9 22

1 log3 27 23

3 log3 81 24

9 log3 243 25

Identify key features.

The asymptote is the vertical line x 5 0, since in this equation, h 5 0.

The parameter b is equal to 27 in this function.

The parameter a is equal to 21 in this function. When the parameter a is negative, you need to reflect the graph across the x-axis. Since |a| 5 1, the graph will not be stretched vertically.

3

1

Plot the points to sketch the graph.

x

y

2–2 4 6 8 100

–2

–4

–6

–8

2

4

y � �log3 27x

2

Starting from the parent graph y 5 log5 x,

what transformations must you make to

graph g(x) 5 2log5 1 ___ 25 (x 1 12)?

DISCUSS


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EXAMPLE A Graph transformations of y 5 log x, which is displayed here, to show that log 10x 5 log x 1 1.

x

y

2–2 4 6 8 10 120

–2

–4

–6

2

4

6

y � log x

Graph y 5 log 10x.

You can think of the equation as y 5 log 10(x 1 0). So, the vertical asymptote is x 5 0.

The coefficient 10 tells you that the graph is

a horizontal shrink of y 5 log x by 10. That

is, you should multiply the x-values of the

ordered pairs by 1 __ 10 . For example, y 5 log x

contains the points (10, 1) and (100, 2).

So, the graph of y 5 log 10x contains the

points (1, 1) and (10, 2).

x

y

2–2 4 6 8 10 120

–2

–4

–6

2

4

6

y � log 10x

Draw a conclusion.

▸The graphs are the same. The expressions log 10x and log x 1 1 are equivalent.

3

1

Graph y 5 log x 1 1.

Graph a vertical translation of y 5 log x up 1 unit.

x

y

2–2 4 6 8 10 120

–2

–4

–6

2

4

6

y � log x � 1

2

Sketch the graph of y 5 log 210x.

TRY


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EXAMPLE B The graph of y 5 log3 x is shown. Graph y 5 log3 (x 2 1) 2 2.

x

y

2–2 4 6 8 10 120

–2

–4

–6

2

4

6

y � log3 x

Determine which transformations you will make to the parent graph.

Compare the equations.

Both logarithms use the same base, 3.

Using the form y 5 a log [b(x 2 h)] 1 k, you find that the value of h in y 5 log3 (x 2 1) 2 2 is 1.

This means the graph, including the asymptote, is translated 1 unit to the right.

For any function f (x), the graph of f (x) 1 k is a translation k units up or down.

Since 2 is subtracted from the logarithmic expression, the graph is translated down 2 units.

1

Graph the function.

Move each point on the parent graph 1 unit right and 2 units down.

▸The graph of y 5 log3 (x 2 1) 2 2 is shown here.

x

y

2–2 4 6 8 10 120

–2

–4

–6

2

4

6

y � log3 (x � 1) � 2

2

Pick several points on the graph, and test them in the function.

CHECK


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EXAMPLE C The graph of y 5 log3 x is shown.

Graph y 5 1 __ 2 log3 x __ 3 .

x

y

2–2 4 6 8 10 120

–2

–4

–6

2

4

6

y � log3 x



Think of y 5 1 __ 2 log3 x __ 3 as y 5 1 __ 2 log3 1 __ 3 x.

Both logarithms use the same base, 3.

Using the form y 5 a log [b(x 2 h)] 1 k, you

find that a 5 1 __ 2 , b 5 1 __ 3 , h 5 0, and k 5 0 in

y 5 1 __ 2 log3 x __ 3 .

The value of a is a vertical shrink of the

graph of y 5 log3 x by 1 __ 2 .

The value of b is a horizontal stretch of the

graph of y 5 log3 x by 1 __ 3 .

Sketch the graph.

Plot the points you found, and draw the graph.

▸ The graph of y 5 1 __ 2 log3 x __ 3 is shown here.

x

y

2–2 4 6 8 10 120

–1

–2

–3

1

2

3

1

Find several points of the function.

Find the x-intercept in the form ( 1 __ b 1 h, 0 ) with b 5 1 __ 3 and h 5 0.

The x-intercept is (3, 0).

Find several points of the parent function.

Then perform the shrink and the stretch.

Multiply the x-value by 3 and the y-value

by 1 __ 2 .

The point (2, 0.6) becomes (6, 0.3).

The point (3, 1) becomes ( 9, 1 __ 2 ) .

2

Describe the transformations required to change the graph of y 5 log3 x into the graph of y 5 2 log3 (4x 1 4).

DISCUSS


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EXAMPLE D The graph of y 5 log2 x is shown. Graph y 5 log2 2(x 1 1).

x

y

–2 2 4 6 8 100

–2

2

4

6

8

10

y � log2 x

Shrink the graph horizontally toward the asymptote.

Since b 5 2, the distance between any point and the vertical asymptote will be halved.

▸

x

y

2–2 4 6 8 100

–2

–4

–6

2

4

6

y � log2 2(x � 1)

4



Using the form y 5 a log [b(x 2 h)] 1 k, you find that a 5 1, b 5 2, h 5 21, and k 5 0 in y 5 log2 2(x 1 1).

Since b 5 2, the graph is shrunk horizontally toward the asymptote by 2.

Since h 5 21, the graph is translated 1 unit to the left, and the vertical asymptote is x 5 21.

Translate the parent graph 1 unit left.

x

y

2–2 4 6 8 100

–2

–4

–6

2

4

6

3

1

Find the intercepts.

The x-intercept is located where x 5 1 __ b 1 h.

1 __ 2 1 (21) 5 2 1 __ 2

The x-intercept is ( 21 __ 2 , 0 ) .

The y-intercept is located where x 5 0.

log2 2(0 1 1) 5 log2 2 5 1

The y-intercept is (0, 1).

2

How can you know if a logarithmic function has a y-intercept?

DISCUSS

Practice


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Complete each table. Then sketch a graph of each function.

1. y 5 log4 x

x y

1 __ 4

1

4

8

REMEMBER logb a 5 log a

____ log b and logb b 5 1

2. y 5 log5 x

x y

1 __ 5

1

5

For each function, identify the features listed, if possible. If none exists, write none.

3. y 5 log6 x 2 5

asymptote:

x-intercept:

y-intercept:

4. y 5 log9 4(x 1 3)

asymptote:

x-intercept:

y-intercept:

REMEMBER A function in the form

y 5 a log [b (x 2 h)] has an x-intercept of ( 1 _ b 1 h, 0 ) .

x

y

2–2–4 4 6 8 100

–1

–2

–3

1

2

3

x

y

2–2–4 4 6 8 100

–1

–2

–3

1

2

3


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On each coordinate plane, the graph of y 5 log2 x is shown. Graph the given function on the same coordinate plane.

Choose the best answer.

7. A function of the form y 5 a log [b(x 2 h)] 1 k is shown on the coordinate plane.

x

y

–2–4–6 2 4 60

–2

–4

2

4

6

–6

Which of the following is true of the function?

A. h 5 24

B. h 5 23

C. h 5 3

D. h 5 4

5. y 5 log2 (x 2 2)

x

y

–2 2 4 6 8 100

–2

2

4

6

8

10

y � log2 x

6. y 5 log2 x 1 2

x

y

–2 2 4 6 8 100

–2

2

4

6

8

10

y � log2 x


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8. y 5 2 log3 x

x

y

2–2 4 6 8 10 120

–2

–4

–6

2

4

6

y � log3 x

9. y 5 log3 x __ 2

x

y

2–2 4 6 8 10 120

–2

–4

–6

2

4

6

y � log3 x


10. y 5 2 log4 (x 2 3)

x

y

2–2 4 6 8 10 120

–1

–2

–3

1

2

3

11. y 5 log4 2x 2 1

x

y

2–2 4 6 8 10 120

–1

–2

–3

1

2

3


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Solve.

12. PREDICT Consider the logarithmic function

y 5 23 log2 21 __ 2 x.

The parent graph, y 5 log2 x, is shown.

Predict how the graph of y 5 23 log2 21 __ 2 x will

transform the parent graph.

Fill in the table, and sketch the graph on the coordinate plane above.

Was your prediction correct?

13. STRATEGIZE Look at the table in the previous question. The x-values used all give y-values that are integers. The table below will be used to find (x, y) ordered pairs for the equation y 5 log6 2x. Come up with a strategy to find x-values that will give integer y-values, and explain why it works. Then fill in the table.

Strategy:

x log6 2x y

x

y

–2–4–6–8–10 2 4 6 8 100

–2

–4

–6

–8

2

4

6

8

x 23 log22 1

__ 2 x y

21 23 log2 1 __ 2

22 23 log2 1

24

28

Exponential and Logarithmic Graphs...Lesson 24: Graphing Logarithmic Functions 229onnect Duplicating...

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Transcript of Exponential and Logarithmic Graphs...Lesson 24: Graphing Logarithmic Functions 229onnect Duplicating...