Section 7.3 Page 378 to 386

Product and Quotient Laws of Logarithms

Recall the exponent laws that you have already learned. How do you think these laws relate to logarithms, given that a logarithm is the inverse of an exponent?

The power law of logarithms introduced in Chapter 6 states that

logb xn � n logb x, b � 0, b � 1, x � 0

What do the laws of logarithms have to do with music?

Investigate What is an equivalent expression for the logarithm

of a product or of a quotient?

A: Common Logarithm of a Product: Make Geometric and Algebraic Connections

1. Evaluate the following.

a) log 1 b) log 10 c) log 100 d) log 1000 e) log 10 000

2. a) Graph the following family of functions on the same grid.

f(x) � log x

g(x) � log (10x)

h(x) � log (100x)

i(x) � log (1000x)

j(x) � log (10 000x)

b) Describe how these graphs are related to each other using vertical translations. To help recognize the translation, consider what happens to the points (1, 0) and (10, 1) in each case.

3. a) Copy and complete the table. Write each function in three different ways, as shown in the fi rst row. The entry in the third column is found by converting the translation coeffi cient into a common logarithm.

b) Reflec t Describe any pattern you observe between the first and last columns.

Tools• computer with The Geometer’s

Sketchpad®or

• graphing calculator

or

• computer algebra system (CAS)

Function Vertical Translation of f(x) Sum of Common Logarithms

g(x) � log (10x) g(x) � log x � 1 g(x) � log x � log 10

h(x) � log (100x)

i(x) � log (1000x)

j(x) � log (10 000x)

7.3

378 MHR • Advanced Functions • Chapter 7

B: Extend the Pattern to Logarithms of Any Base

Does the pattern observed in part A apply to logarithms of any base?

1. Copy and complete the table.

2. Look at the pattern of results in the table. What does this suggest about the base-2 logarithm of a product?

3. Make a conjecture about how a logarithm of a product with any base can be written in terms of a sum.

4. Repeat steps 1 and 2 for logarithms of various bases. Compare your results to your prediction in step 3, and determine whether your prediction is correct.

5. Ref lec t

a) Is there a product law for logarithms? Write it down, using

i) algebraic symbols

ii) words

b) Are there any restrictions or special considerations? If so, state them using

i) algebraic symbols

ii) words

C: Explore the Common Logarithm of a Quotient

1. Is there a quotient law for logarithms? Design and carry out an investigation that addresses this question using tools and strategies of your choice. Carry out the investigation.

2. Ref lec t Compare your results with those of your classmates. Write a brief report of your fi ndings that includes

• a quotient law for logarithms, written using words and symbols

• three examples that illustrate the law

• the values of the base for which the law holds true

Technology Tip s

Spreadsheets, computer algebra

systems, and graphing calculators

are all useful tools for performing

repeated calculations.

a b log2 a log

2 b log

2 a � log

2 b log

2 (a � b)

1 2 log2 1 � 0 log2 2 � 1 0 � 1 � 1 log2 (1 � 2)� log2 2� 1

2 4

4 4

8 16

Create at least two of your own examples.

7.3 Product and Quotient Laws of Logarithms • MHR 379

The product law of logarithms can be developed algebraically as follows.

Let x � logb m and y � logb n.

Write these equations in exponential form:

bx � m and by � n

mn � bxby

mn � bx � y Use the product law of exponents.

logb mn � x � y Write in logarithmic form.

logb mn � logb m � logb n Substitute for x and y.

You will prove the quotient law in question 20.

Product Law of Logarithms

logb (mn) � logb m � logb n for b � 0, b � 1, m � 0, n � 0.

Quotient Law of Logarithms

logb ( m _ n ) = logb m � logb n for b � 0, b � 1, m � 0, n � 0.

The product and quotient laws of logarithms, as well as the power law, are useful tools for simplifying algebraic expressions and solving equations.

Example 1 Simplify by Applying Laws of Logarithms

Write as a single logarithm.

a) log5 6 � log5 8 � log5 16

b) log x � log y � log (3x) � log y

Solution

a) log5 6 � log5 8 � log5 16 � log5 ( 6 � 8 __ 16

) Apply the product and quotient laws of logarithms.

� log5 3

b) log x � log y � log (3x) � log y

Method 1: Apply Laws of Logarithms Directly

log x � log y � log (3x) � log y � log ( xy(3x)

__ y ) Apply the product and quotient

laws of logarithms.

� log (3x2), x � 0, y � 0

The power law of logarithms cannot be applied here, because the exponent 2 applies only to x, not to 3x. Therefore, this is the simplest form of the expression.


Note the restrictions that must be placed on the variables, based on the nature of the original expression log x � log y � log (3x) � log y. The simplifi ed expression is only defi ned for values of x and y for which each logarithmic term in the original expression is defi ned.

Method 2: Simplify First, and Then Apply the Product Law

log x � log y � log (3x) � log y � log x � log (3x) Collect like terms and simplify.

� log [(x)(3x)] Apply the product law of logarithms.

� log (3x2), x � 0, y � 0

Example 2 Evaluate Using the Laws of Logarithms

Evaluate.

a) log8 4 � log8 16

b) log3 405 � log3 5

c) 2 log 5 � 1 _ 2 log 16

Solution

a) log8 4 � log8 16 � log8 (4 � 16) Apply the product law of logarithms.

� log8 64

� 2 82 � 64

b) log3 405 � log3 5 � log3 ( 405 _ 5 ) Apply the quotient law of logarithms.

� log3 81

� 4

c) 2 log 5 � 1 _ 2 log 16 � log 52 � log 1 6

1 _ 2 Apply the power law of logarithms.

� log 25 � log √ �� 16

� log 25 � log 4

� log (25 � 4) Apply the product law of logarithms.

� log 100

� 2


Example 3 Write the Logarithm of a Product or Quotient

as a Sum or Diff erence of Logarithms

Write as a sum or difference of logarithms. Simplify, if possible.

a) log3 (xy) b) log 20 c) log (ab2c) d) log uv _ √ � w

Solution

a) log3 (xy) � log3 x � log3 y, x � 0, y � 0 Apply the product law of logarithms.

b) Method 1: Use Factors 4 and 5 Method 2: Use Factors 2 and 10 log 20 � log 4 � log 5 log 20 � log 10 � log 2

� 1 � log 2

c) log (ab2c) � log a � log b2 � log c

� log a � 2 log b � log c, a � 0, b � 0, c � 0 Apply the power law of logarithms.

d) log ( uv _ √ � w

) � log u � log v � log √ � w Apply the product and quotient laws of logarithms.

� log u � log v � log w 1 _ 2

Write the radical

in exponential form.

� log u � log v � 1 _ 2 log w, u � 0, v � 0, w � 0 Apply the power

law of logarithms.

Example 4 Simplify Algebraic Expressions

Simplify.

a) log ( √ � x

_ x2

)

b) log ( √ � x )3 � log x2 � log √ � x

c) log (2x � 2) � log (x2 � 1)

Solution

a) Method 1: Apply the Quotient Law of Logarithms

log ( √ � x

_ x2

) � log √ � x � log x2 Apply the quotient law of logarithms.

� log x 1 _ 2

� log x2 Write the radical in exponential form.

� 1 _ 2 log x � 2 log x Apply the power law of logarithms.

� 1 _ 2 log x � 4 _

2 log x

� � 3 _ 2 log x, x � 0 Collect like terms.


Method 2: Apply the Quotient Law of Exponents

log ( √ � x

_ x2

) � log ( x 1 _ 2

_

x2 ) Write the radical in exponential form.

� log x 1 _ 2

�2 Apply the quotient law of exponents.

� log x � 3 _

2

� � 3 _ 2 log x, x � 0 Apply the power law of logarithms.

b) log( √ � x )3 � log x2 � log √ � x � log x 3 _ 2

� log x2 � log x

1 _ 2

� log ( ( x

3 _ 2

) (x2) __

( x 1 _ 2 )

) Apply the product and quotient laws of logarithms.

� log x 3 _ 2

� 2 � 1 _ 2 Apply the product and quotient

laws of exponents. � log x3

� 3 log x, x � 0 Apply the power law of logarithms.

c) log(2x � 2) � log(x2 � 1) � log ( 2x � 2 __ x2 � 1

)

� log ( 2(x � 1)

___ (x � 1)(x � 1)

) Factor the numerator

and denominator.

� log ( 2 __ x � 1

) Simplify by dividing common factors.

To determine the restrictions on the variable x, consider the original expression:

log (2x � 2) � log (x2 � 1)

For this expression to be defi ned, both logarithmic terms must be defi ned. This implies the following:

2x � 2 � 0 x2 � 1 � 0 2x � 2 x2 � 1 x � 1 x � 1 or x � �1

For both logarithmic terms to be defi ned, therefore, the restriction x � 1 must be imposed.

log (2x � 2) � log (x2 � 1) � log ( 2 __ x � 1

) , x � 1

Note that this restriction ensures that all logarithmic expressions in the equation are defi ned.

C O N N E C T I O N S

A more concise way to write

x > 1 or x < �1 is to use

absolute value notation: �x� > 1.


<< >>KEY CONCEPTS

The product law of logarithms states that logb x � logb y � logb (xy) for b � 0, b � 1, x � 0, y � 0.

The quotient law of logarithms states that logb x � logb y � logb ( x _ y ) for b � 0, b � 1, x � 0, y � 0.

The laws of logarithms can be used to simplify expressions and solve equations.

Communicate Your Understanding

C1 Does log x � log y � log (x � y)? If so, prove it. If not, explain why not. Use examples to support your answer.

C2 Does log ( a _ b ) �

log a _

log b ? If so, prove it. If not, explain why not.

Use examples to support your answer.

C3 Refer to the fi nal line in the solution to Example 4. Why is the restriction on the variable x � 1 and not �x� � 1?

C4 Summarize the laws of logarithms in your notebook. Create an example to illustrate each law.

C5 Refl ect on the domain and range of y � log x. Why must x be a positive number?

A Practise

For help with questions 1 to 3, refer to Example 1.

1. Simplify, using the laws of logarithms.

a) log 9 � log 6

b) log 48 � log 6

c) log3 7 � log3 3

d) log5 36 � log5 18

2. Use Technology Use a calculator to evaluate each result in question 1, correct to three decimal places.

3. Simplify each algebraic expression. State any restrictions on the variables.

a) log x � log y � log (2z)

b) log2 a � log2 (3b) � log2 (2c)

c) 2 log m � 3 log n � 4 log y

d) 2 log u � log v � 1 _ 2 log w

For help with questions 4 to 6, refer to Example 2.

4. Evaluate, using the product law of logarithms.

a) log6 18 � log6 2

b) log 40 � log 2.5

c) log12 8 � log12 2 � log12 9

d) log 5 � log 40 � log 5

5. Evaluate, using the quotient law of logarithms.

a) log3 54 � log3 2

b) log 50 000 � log 5

c) log4 320 � log4 5

d) log 2 � log 200

6. Evaluate, using the laws of logarithms.

a) 3 log16 2 � 2 log16 8 � log16 2

b) log 20 � log 2 � 1 _ 3 log 125


B Connect and Apply

For help with questions 9 and 10, refer to Example 4.

9. Simplify. State any restrictions on the variables.

a) log ( x2

_ √ � x

)

b) log ( √ � m _ m3

) � log( √ � m )7

c) log √ � k � log ( √ � k )3 � log 3 √ � k2

d) 2 log w � 3 log √ � w � 1 _ 2 log w2

10. Simplify. State any restrictions on the variables.

a) log (x2 � 4) � log (x � 2)

b) log (x2 � 7x � 12) � log (x � 3)

c) log (x2 � x � 6) � log (2x � 6)

d) log (x2 � 7x � 12) � log (x2 � 9)

11. Chapter Problem A certain operational amplifi er (Op Amp) produces a voltage output, V0, in volts, from two input voltage signals, V1 and V2, according to the equation V0 � log V2 � log V1.

a) Write a simplifi ed form of this formula, expressing the right side as a single logarithm.

b) What is the voltage output if

i) V2 is 10 times V1?

ii) V2 is 100 times V1?

iii) V2 is equal to V1?

12. a) Explain how you can transform the graph of f(x) � log x to produce g(x) � log (10nx), for any n � 0.

b) Create two examples to support your explanation. Sketch graphs to illustrate.

13. Refer to question 12 part a). Can this process be applied when n is an integer? Explain.

14. Use Technology

a) Graph the functions f(x) � 2 log x and g(x) � 3 log x.

b) Graph the sum of these two functions: p(x) � f(x) � g(x).

c) Graph the function q(x) � log x5.

d) How are the functions p(x) and q(x) related? What law of logarithms does this illustrate?


Op Amps are tiny integrated circuits that can be used to perform various

mathematical functions on voltage signals. Engineers combine Op

Amps with other electronic circuit elements to create useful equipment

such as electric guitar amplifi ers, eff ects pedals, and signal processors.

For help with questions 7 and 8, refer to Example 3.

7. Write as a sum or difference of logarithms. Simplify, if possible.

a) log7 (cd) b) log3 ( m _ n )

c) log (uv3) d) log ( a √ � b _

c2 )

e) log2 10 f) log5 50

8. Refl ect on your answer to question 7f). Is there more than one possibility? Explain. Which solution gives the simplest result?

Connecting

Problem Solving

Reasoning and Proving

Reflecting

Selecting ToolsRepresenting

Communicating


You will learn more about the sums of functions in Chapter 8.

Technology Tip s

To distinguish two graphs on a graphing calculator screen, you can

alter the line style. Cursor over to the left of Y2� and press e.

Observe the diff erent line styles available. Press e to select a style.

You can change colours and line thicknesses in The Geometer’s Sketchpad® by right-clicking on the curve.


15. Use Technology Refer to question 14.

a) In place of g(x) � 3 log x use g(x) � 4 log x.

How should the function q(x), in step c), change in order to illustrate the same property as in question 14?

b) Graph the functions p(x) and q(x).

c) You should observe something unusual about the graph of q(x). Explain what is unusual about this graph and where the unusual portion comes from.

16. Use Technology Create a graphical example that illustrates the quotient law of logarithms using a graphing calculator with CAS or other graphing technology. Explain how your example works.

17. Use the power law of logarithms to verify the product law of logarithms for log 102.

18. Renata, a recent business school graduate, has been offered entry-level positions with two fi rms. Firm A offers a starting salary of $40 000 per year, with a $2000 per year increase guaranteed each subsequent year. Firm B offers a starting salary of $38 500, with a 5% increase every year after that.

a) After how many years will Renata earn the same amount at either fi rm?

b) At which fi rm will Renata earn more after

i) 10 years? ii) 20 years?

c) What other factors might affect Renata’s choice, such as opportunities for promotion? Explain how these factors may infl uence her decision.

✓ Achievement Check

19. Express each as a single logarithm. Then, evaluate, if possible.

a) log4 192 � log4 3

b) log5 35 � log5 7 � log5 25

c) loga (ab) � loga (a3b)

d) log (xy) � log y _ x

C Extend and Challenge

20. Prove the quotient law of logarithms by applying algebraic reasoning.

21. a) Given the product law of logarithms, prove the product law of exponents.

b) Given the quotient law of logarithms, prove the quotient law of exponents.

22. Math Contest For what positive value of c does the line y � �x � c intersect the circle x2 � y2 � 1 at exactly one point?

23. Math Contest For what value of k does the parabola y � x2 � k intersect the circle x2 � y2 � 1 in exactly three points?

24. Math Contest A line, l1, has slope �2 and passes through the point (r, �3). A second line, l2, is perpendicular to l1, intersects l1 at the point (a, b), and passes through the point (6, r). What is the value of a?

A r B 2 _ 5 r C 1 D 2r � 3 E 5 _

2 r

25. Math Contest A grocer has c pounds of coffee divided equally among k sacks. She fi nds n empty sacks and decides to redistribute the coffee equally among the k � n sacks. When this is done, how many fewer pounds of coffee does each of the original sacks hold?

Connecting

Problem Solving

Reasoning and Proving

Reflecting

Selecting ToolsRepresenting

Communicating


Documents

Section 7.3 Page 378 to 386