Section 6.2 Page 323 to 330

Logarithms6.2

Suppose a new Web site becomes very popular very quickly. The number of visitors as a function of time can be modelled using an exponential function. What if you wanted to describe the time required for the number of visitors to reach a certain value? This type of functional relationship can be modelled using the inverse of an exponential function.

y

x

y � bx

x � by

y � x

0

y

x

y � bx

y � logbx

y � x

0

Because each x-value in the inverse graph gives a unique y-value, the inverse is also a function of x. In the inverse function, the y-value is the exponent to which the base, b, must be raised to produce x, by � x. In the mathematics of functions, we usually prefer to express the y-coordinate in terms of the x-coordinate, so we restate the relationship as y being the logarithm of x to the base b. This relationship is written as y � logb x.

Any exponential relationship can be written using logarithm notation:

23 � 8 ↔ log2 8 � 3

52 � 25 ↔ log5 25 � 2

r s � t ↔ logr t � s

The logarithmic function is defi ned as y � logb x, or y equals the logarithm of x to the base b.

This function is defi ned only for b � 0, b � 1.

Using this notation, the logarithm, y, is the exponent to which the base, b, must be raised to give the value x. The logarithmic function is the inverse of the exponential function with the same base. Therefore, any equation of the form y � bx can be written in logarithmic form.

6.2 Logarithms • MHR 323

Example 1 Write an Exponential Equation in Logarithmic Form

Rewrite each equation in logarithmic form.

a) 16 � 24

b) m � n3

c) 3�2 � 1 _ 9

Solution

a) 16 � 24

4 � log2 16

b) m � n3

logn m � 3

c) 3�2 � 1 _ 9

log3 ( 1 _ 9 ) � �2

Note that logarithms can produce negative results, as in part c) above, but the base of a logarithm can never be negative, zero, or one. Why is this? A negative base with a non-integer exponent is undefi ned, log0 0 has an infi nite number of solutions, and log1 x only has meaning for x � 1, in which case it has an infi nite number of values.

The logarithmic function is useful for solving for unknown exponents.

Example 2 Evaluate a Logarithm

Evaluate.

a) log3 81

b) log2 ( 1 _ 8 )

c) log10 0.01

Solution

a) The logarithm is the exponent to which you must raise a base to produce a given value.

Let y � log3 81

Then, 3y � 81 3y � 34

y � 4

C O N N E C T I O N S

4 � log2 16 is read as “4 equals

the logarithm of 16 to the base 2.”

Notice that in both forms of the

equation, the base is 2:

16 � 24 4 � log2 16

base

324 MHR • Advanced Functions • Chapter 6

b) log2 ( 1 _ 8 )

Method 1: Mental Calculation

1 _ 8 � 1 _

23

� 2�3 Think: To what exponent must 2 be raised to produce the value 1 _ 8

?

log2 ( 1 _ 8 ) � �3

c) Let y � log10 0.01.

Then 10y � 0.01 10y � 10�2 y � �2

Method 2: Graphical Analysis

Graph the function y � 2x, and fi nd the value of x that corresponds to

y � 1 _ 8 or 0.125. A graphing calculator can be used for this.

Notice from Example 2a) and c) that loga (ab) � b.

Logarithms to the base 10 are called common logarithms . When writing a common logarithm, it is not necessary to write the base; that is, log 100 is understood to mean the same as log10 100.

Example 3 Write a Logarithmic Equation in Exponential Form

Rewrite in exponential form.

a) log4 64 � 3 b) y � log x

Solution

a) log4 64 � 3

The base is 4 and the exponent is 3. In exponential form, this equation is 43 � 64.

b) y � log x

Because there is no base written, this function is understood to be the common logarithm of x. In exponential form, this is 10y � x.

Technology Tip s

You can see from the graph in

the standard viewing window

that y � 0.125 at some point

when x � 0. Adjust the viewing

window and zoom in to fi nd the

point where y � 0.125.


Example 4 Approximate Logarithms

Find an approximate value for each logarithm.

a) log2 10 b) log 2500

Solution

a) log2 10

Method 1: Systematic Triallog2 8 � 3 and log2 16 � 4, so log2 10 must be between 3 and 4. Find the approximate exponent to which 2 must be raised to give 10. Try 3.50 fi rst.

Therefore, log2 10 � 3.32.

Method 2: Graphical Analysis of y � 2x

Trace the graph of y � 2x and fi nd the value of x that produces y � 10. Graphing software can be used to do this.

The graph shows that y � 10 when x � 3.32, so log2 10 � 3.32.

Method 3: Intersection of Two FunctionsTo fi nd the value of x when 2x � 10, enter the left side and the right side into a graphing calculator, each as a separate function. Then, fi nd their point of intersection.

These graphs intersect when x � 3.32, so log2 10 � 3.32.

Estimate Check Analysis

3.50 23.50 � 11.3 Too high. Try a lower value.3.10 23.10 � 8.6 Too low. Try 3.3.

3.30 23.30 � 9.8 Low, but very close.3.35 23.35 � 10.2 A little high.3.32 23.32 � 10.0 This is a good estimate.

Technology Tip s

To fi nd the intersection point

using a graphing calculator,

press O r for [CALC].

Choose 5:intersect. Press

e three times.

Connecting

Problem Solving

Reasoning and Proving

Reflecting

Selecting ToolsRepresenting

Communicating


<< >>

b) log 2500

While any of the methods illustrated above can be used here, a scientifi c or graphing calculator can be used to calculate common logarithms.

log 2500 � 3.40

Later in this chapter, you will analyse a technique for evaluating logarithms with any base, using a calculator.

KEY CONCEPTS

The logarithmic function is the inverse of the exponential function.

The value of logb x is equal to the exponent to which the base, b, is raised to produce x.

Exponential equations can be written in logarithmic form, and vice versa.

y � bx ↔ x � logb y

y � logb x ↔ x � by

Exponential and logarithmic functions are defi ned only for positive values of the base that are not equal to one:

y � bx, b � 0, x � 0, b � 1

y � logb x, b � 0, y � 0, b � 1

The logarithm of x to base 1 is only valid when x � 1, in which case y has an infi nite number of solutions and is not a function.

Common logarithms are logarithms with a base of 10. It is not necessary to write the base for common logarithms: log x means the same as log10 x.

Communicate Your Understanding

C1 Is a logarithm an exponent? Explain.

C2 Consider an equation in logarithmic form. Identify each value in the equation and describe what it means. Discuss how this equation would appear in exponential form.

C3 Does log2 (�4) have meaning? If so, explain what it means. If not, explain why it has no meaning.

y

x

exponential functionlogarithmic function

0


A Practise

For help with question 1, refer to Example 1.

1. Rewrite each equation in logarithmic form.

a) 43 � 64

b) 128 � 27

c) 5�2 � 1 _ 25

d) ( 1 _ 2 )

2

� 0.25

e) 6x � y

f) 105 � 100 000

g) 1 _ 27

� 3�3

h) v � bu

For help with questions 2 and 3, refer to Example 2.

2. Evaluate each logarithm.

a) log2 64

b) log3 27

c) log2 ( 1 _ 4 )

d) log4 ( 1 _ 64

) e) log5 125

f) log2 1024

g) log6 363

h) log3 81

3. Evaluate each common logarithm.

a) log 1000

b) log ( 1 _ 10

) c) log 1

d) log 0.001

e) log 10�4

f) log 1 000 000

g) log ( 1 _ 100

) h) log 10 000

For help with question 4, refer to Example 3.

4. Rewrite in exponential form.

a) log7 49 � 2

b) 5 � log2 32

c) log 10 000 � 4

d) w � logb z

e) log2 8 � 3

f) log5 625 � 4

g) �2 � log ( 1 _ 100

) h) log7 x � 2y

5. Sketch a graph of each function. Then, sketch a graph of the inverse of each function. Label each graph with its equation.

a) y � 2x

b) y � 4x

For help with questions 6 to 8, refer to Example 4.

6. Estimate the value of each logarithm, correct to one decimal place, using a graphical method.

a) log2 6

b) log4 180

c) log3 900

d) log9 0.035

7. Pick one part from question 6. Use a different graphical method to verify your answer.

8. Evaluate, correct to two decimal places, using a calculator.

a) log 425

b) log 0.000 037

c) log 9

d) log 0.2

e) log 17

f) log 99

g) log 183

h) log 1010


B Connect and Apply

9. Let y � log x.a) Write the corresponding inverse function

in exponential form.b) Sketch a graph of y � log x and its inverse

on the same grid.

10. Evaluate each logarithm.

a) log3 3 b) log2 2

c) log12 12 d) lo g 1 _ 2 ( 1 _

2 )

11. a) Make a prediction about the value of logx x for any value of x � 0, x � 1.

b) Test your prediction by evaluating several cases.

c) What can you conclude about the value of logx x? Explain your answer using algebraic, numerical, or graphical reasoning.

12. a) Compare the rate of change of a logarithmic function to that of its inverse (exponential) function.

b) How are their rates of change different? How are they alike? Use an example to illustrate and support your explanation.

13. The number of visitors to a popular Web site is tripling every day. The time, t, in days for a number, N, of visitors to see the site

is given by the equation t � log N

_ log 3

.

How long will it take until the number of visitors to the Web site reaches each number?

a) 1000 b) 1 000 000

14. Fog can greatly reduce the intensity of oncoming headlights. The distance, d, in metres, of an oncoming car whose headlights have an intensity of light, I, in lumens (lm),

is given by d � �167 log ( I _ 125

) .a) How far away is a car whose headlight

intensity is 50 lm?b) If the headlight intensity doubles, does this

mean the car is half as far away? Explain.c) What implications do these results have on

recommended driver behaviour?

15. Chapter Problem Engineers of spacecraft take care to ensure that the crew is safe from the dangerous cosmic radiation of space. The protective hull of the ship is constructed of a special alloy that blocks radiation according to the equation P � 100(0.2)x, where P is the percent of radiation transmitted through a hull with a thickness of x centimetres.

a) What thickness of the hull walls will ensure that less than 1% of the radiation will pass through?

b) By how much would the thickness of the walls need to be increased in order to reduce the harmful radiation transmission to 0.1%?

✓Achievement Check

16. After hearing some mysterious scratching noises, four friends at a high school decide to spread a rumour that there are two hedgehogs living inside the school’s walls. Each friend agrees to tell the rumour to two other students every day, and also to encourage them to do likewise. Assuming that no one hears the rumour twice, the time, t, in days, it will take for the rumour to reach N students

is given by t � log ( N _

4 ) __

log 3 .

a) Determine how long it will take for the rumour to reach

i) 30 students

ii) half of the student population of 1100

iii) the entire student population

b) Graph the function.

c) Describe how the graph would change if the number of students who initially began the rumour were

i) greater than 4

ii) less than 4

Explain your reasoning.

d) Describe how the graph would change if some students were to hear the rumour more than once. Explain your reasoning.

Connecting

Problem Solving

Reasoning and Proving

Reflecting

Selecting ToolsRepresenting

Communicating


C Extend and Challenge

17. Use Technology Graph the functiony � log x using a graphing calculator or graphing software. Experiment with the Zoom and Window settings. Try to view the function over as large a range as possible.

a) Approximately how many integer values of y can be viewed at any one time?

b) Explain why it is diffi cult to view a broad range of this function.

18. Because logarithmic functions grow very slowly, it is diffi cult to see much of their range using normal graphing methods. One method of getting around this is to use semi-log graph paper, in which one variable is graphed versus the common logarithm of the other variable.

a) Create a table of values for the function y � log x, using several powers of 10 for x.

b) Graph the function on semi-log graph paper. Describe the shape of the graph. Explain why it has this shape.

c) What are some advantages of using this technique to graph logarithmic relationships?

19. Use Technology Another technique that can be used to view logarithmic functions over a broad range is to linearize the relationship using spreadsheet software.

a) Enter the table of values from question 18 for the function y � log x. Use column A for x and column B for y, starting at row 1.

b) Create a scatter plot of the data. Describe the shape of the graph of y � log x.

c) Use the formula � log(A1) in cell C1 and copy it to cells C2 to C4 to fi nd the common logarithm of column A.

d) Create a scatter plot using the data in columns C and B. Describe the shape of this graph.

e) What are some advantages of using this technique to graph a logarithmic relationship?

20. Math Contest The ratio 102006 � 102008

___ 102007 � 102007

is

closest to which of the following numbers?

A 0.1 B 0.2 C 1 D 5 E 10

21. Math Contest Two different positive numbers, m and n, each differ from their reciprocals by 1. What is the value of m � n?

0

1

0

2

3

4

5

6

7

8

1 10 100 1000 10 000 100 000 1 000 000

y

x

C O N N E C T I O N S

In laboratory experiments, semi-logarithmic scatter plots can

be used to determine an unknown parameter that occurs in the

exponent of an exponential relationship. You will learn more

about semi-log plots if you study physics or chemistry

at university.

Log charts are also used in the stock market to track a stock’s

value over a period of time.


Documents

Section 6.2 Page 323 to 330