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430 MHR Chapter 7
Investigate & Inquire: Laws of Logarithms1. a) Copy and complete the table.
b) Examine the results of each row. Make a conjecture about the product law forlogarithms.c) Test your conjecture by evaluating log6 (36 � 216), and make any necessary adjustmentsto your original conjecture.
2. Make and test a conjecture about the power law of logarithms: loga pc � c loga p.
3. a) Copy and complete the table.
b) Examine the results of each row. Make a conjecture about the quotient law forlogarithms.c) Test your conjecture by evaluating , and make any necessary adjustments to
your original conjecture.
log324327
Laws of Logarithms
With the properties of logarithms learned in Section 7.2, we can work with only a limitednumber of logarithmic situations. There are other properties of logarithms that are usefulfor solving exponential and logarithmic equations. Such equations arise in a variety ofcontexts, such as investments and bacterial growth.
In the following investigation, you will explore three very important properties oflogarithms, dealing with products, powers, and quotients.
loga x loga y logaxy
log2 32 � log2 8 � log2328
=
log3 27 � log3 9 � log3279
=
log2 32 � log2 16 � log23216
=
log4 64 � log4 16 � log46416
=
log5 25 � log5 5 � log5255
=
log5 25 � log5 5 � log5 125 �
log4 64 � log4 16 � log4 1024 �
log2 16 � log2 32 � log2 512 �
log3 9 � log3 27 � log3 243 �
log2 4 �
loga x
log2 8 �
loga y
log2 32 �
loga (xy)
7.37.3
7.3 Laws of Logarithms MHR 431
d) Remember, log x means log10 x.log 2 � log 50 � log (2)(50) (product law)
� log 100� log 102
� 2
The patterns in the investigation show three laws of logarithms.
Product law: loga (pq) � loga p � loga q
Power law: loga (pc) � c loga p
Quotient law: loga � loga p loga q
Because logarithms can be written as exponents, the laws of exponents can be used tojustify corresponding laws of logarithms. To show the product law, we let loga p � X andloga q � Y. Then, rewriting in exponential form,
loga p � X becomes aX � p andloga q � Y becomes aY � qSo,loga (pq) � loga (aXaY) (substitution)
� loga (aX�Y) (exponents law)� X � Y (loga ax � x)
Substituting for X and Y, loga (pq) � loga p � loga q.
In questions 13 and 14 on page 435, you will show how to derive the power law forlogarithms and the quotient law for logarithms.Since logarithms with base 10 are very common, log10 x is usually written as log x. As wementioned in Section 7.2, the � key on a calculator determines logarithms to base 10.
Example 1 The Laws of Logarithms
Evaluate each expression using the laws of logarithms.
a) log6 4 � log6 9 b) 2 log9 3 c) log3 324 log3 4
d) log 2 � log 50 e) log2 112 log2 7 f )
Solutiona) log6 4 � log6 9 � log6 (4 � 9) (product law)
� log6 36� log6 (62)� 2
c) log log log
log
3 3 3
3
324 4 3244
81
4
�
�
�
log33 9
pq
Web ConnectionFor a visual explanation of the product lawof logarithms, go towww.mcgrawhill.ca/links/CAF12and follow the link.
b) 2 log9 3 � log9 (32) (power law)� log9 9� 1
This expression can be evaluatedwithout using the power law.
2log9 3 � 2 log9
�
��1
2 12
���
�
912
432 MHR Chapter 7
e)
Example 2 Using the Laws of Logarithms
Express as a single logarithm.
a) log7 30 log7 10 b) c) log3 (x2 1) log (x � 1)
Solution
a)
b) Remember, log x means log10 x.
c)
Example 3 Using the Laws of Logarithms
Expand.a) log6 (x2y3) b)
Solutiona) log6 (x2y3) � log6 x2 � log6 y3
� 2 log6 x � 3 log6 y
Recall Example 4 in Section 7.2 (page 426), which we could solve only by graphing becausethe calculator � key determines logarithms only to base 10. With a simple formula, wecan change the base of the logarithm, and use the calculator to evaluate it. For example,
log2
4 abc
log ( ) log ( ) log ( )
log ( )( )
log (
32
3 3
2
3
3
1 1 111 1
1
x xxxx x
x
− − + = −+
= + −+
= xx x− >1 1),
log log log log log log
log ( )( )
12 12
7 2 12 7 2
12 72
12� � �
�
(power law)
product and quotient laws)(
log� 6 7
log log log
log
7 7 7
7
30 10 30103
�
�
(quotient law)
log log log12 12
7 2+ −
log log log
log
2 2 2
2
112 7 1127
16
4
�
�
�
(quotient law) f ) log log (
log
33
3
13
3
9 9
13
9
23
�
�
�
)
(power law)
b) log log log ( )
log (log log )
log log
2
4
24
2
2
14
2 2
214
abc
a bc
a b c
a
= −
= − +
= − 22 2b c− log
7.3 Laws of Logarithms MHR 433
suppose we want to evaluate log2 11 on a calculator. To change the base to 10, we startwith y � log2 11 and proceed as follows.
y � log2 11Write in exponential form. 2y � 11Take the logarithm to base 10 of each side of the equation. log10 (2y) � log10 11Use the power law. y log10 2 � log10 11
Solve for y.
Thus, .
We can now evaluate the logarithm on a calculator.
The general formula for converting a logarithm from one base to another, called the
change of base formula, is
Example 4 Using the Change of Base Formula
Write each logarithm with base 10, and then evaluate it on a calculator. Round your resultsto four decimal places.a) log5 14 b)
Solution
a) b)
Example 5 Doubling Time for an Investment
How long does it take for an investment of $500 to double at 7% interest, compoundedannually?
SolutionThe amount, A, in dollars, of an investment of $500 at 7% interest, compounded annually,for t years, is A � 500(1.07)t. For the investment to double, A � $1000. Thus,1000 � 500(1.07)t
2 � (1.07)t
t � log1.07 2Use the change of base formula to change to base 10.
It takes approximately 10 years and 3 months for an investment of $500 to double at 7%interest, compounded annually.
t = loglog .
.
21 07
10 2448�
loglog
log
.
13
10
10
7713
1 7712
�
�
logloglog
.
510
10
14145
1 6397
=
�
log13
7
logloglog
.ba
a
xxb
�
logloglog2
10
10
11112
=
y =loglog
10
10
112
434 MHR Chapter 7
Key Concepts� For a � (1, �), p, q � (0, �), c � ( �, �),
a) Product Law: loga (pq) � loga p � loga q
b) Quotient Law:
c) Power Law: loga (pc) � c loga p.
� log x means log10 x.
� To change the base of a logarithm from b to a, use the change of base formula
.
Communicate Your Understanding1. Is it possible to use the quotient law of logarithms to evaluate ? Explain.
2. Is it possible to use the product law of logarithms to evaluate log2 7 � log3 8? Explain.3. Explain why the base must be changed to evaluate log5 11 using a calculator.4. Is there more than one way to evaluate log3 9 � log3 3? Explain.5. Is log3 5 equal to log5 3? Explain.
log 74
logloglogb
a
a
xxb
=
log log loga a a
pq
p q�
Practise1. Copy and complete the table.
2. Rewrite each expression using the power law.
a) b) log6 93
c) d) log2 7 5
e) f )
3. Express as a single logarithm.a) log3 5 � log3 8 � log315b) log4 8 log4 10 � log4 3c) log2 19 � log2 4 log2 31
d)
e) log (a � b) � log (a3)f ) log (x � y) log (x y)g) 4 log x 3 log yh) log3 ab � log3 bc
4. Rewrite each expression with no logarithmsof products, quotients, or powers.a) log7 (5x) b) log2 (m3n2)
c) log3 (abc) d)
e) f ) log6 (xy)5
g) h)
5. Evaluate.a) log8 32 � log8 2 b) log2 72 log2 9c) log4 192 log4 3 d) log12 9 � log12 16e) log2 6 � log2 8 log2 3 f ) log3 108 log3 4g) log8 6 log8 3 � log8 4 h) log2 80 log2 5i) log 1.25 � log 80 j) log2 827
log31jk
log4
2a bc
log8
4 mn
log ( )923 y y+
12
17 5log log
log9113
log6 22
15
188log
12
54log
log log12 1253
4�
Single LogarithmSum or Difference of
Logarithmslog2 (12 � 5)
log4 2 � log4 11
log6 (kg)
log8 14 log8 3
log13
2hf
log3 � log3 5
log10 1 log10 7
2 log11 x � 6 log11 x
A
7.3 Laws of Logarithms MHR 435
6. Evaluate to four decimal places using acalculator.a) log5 12 b) log2 13 c) log7 9d) log6 15 e) log9 8 f ) log3 6g) log4 7 h) log8 4
Apply, Solve, Communicate7. Application Driving in fog at night greatlyreduces the intensity of light from anapproaching car. The relationship between thedistance, d, in metres, that your car is from theapproaching car and the intensity of light, I(d),in lumens (lm), at distance d, is given by
a) Solve the equation for I(d).b) How far away from you is an approachingcar if I(d) � 40 lm?
8. Inquiry/Problem Solving Energy is needed totransport a substance from outside a living cellto inside the cell. This energy is measured inkilocalories per gram molecule, and is given by
the relationship , where C1
represents the concentration of the substanceoutside the cell, and C2 represents theconcentration inside the cell.a) Find the energy needed to transport theexterior substance into the cell if theconcentration of the substance inside the cell isi) double the concentration outside the cellii) triple the concentration outside the cellb) What is the sign of E if C1 � C2? Explainwhat this means in terms of the cell.
9. Communication Which is greater, log6 7 orlog8 9? Explain.
10. The formula for the gain in voltage of anelectronic device is Av � 20(log Vo log Vi),where Vo is the output voltage and Vi is theinput voltage.a) Rewrite the formula as a single logarithm.b) Verify the gain in voltage for Vo � 22.8 andVi � 14 using both versions of the formula.
11. When a rope is wrapped around a fixedcircular object, the relationship between the
larger tension TL and the smaller tension TS is
modelled by , where � is the
friction coefficient and � is the wrap angle inradians.
a) Rewrite the formula using the laws oflogarithms.b) If the wrap angle is � (in radians), and a 200 N force is balancing a 250 N force, what is the friction coefficient?c) If the rope is wrapped around the object 2.5 times, what force is now needed to balancethe 250 N force?
12. Show that if logb a � c and logy b � c, thenloga y � c 2.
13. Use the product law of logarithms to provethe quotient law of logarithms,
where a � (1, �), p, q � (0, �).
14. Use the product and quotient laws oflogarithms to prove the power law of logarithmsloga (pc) � c loga p, where a � (1, �), p,q � (0, �), c � ( �, �).
15. Derive the change of base formula,
.
16. Find the error in the following calculation.log3 0.1 � 2 log3 0.1
� log3 (0.1)2
� log3 0.01log3 0.1 � log3 0.01Thus, 0.1 � 0.01.
loglogloga
b
b
xxa
=
log log loga a a
pq
p q�
�
TL TS
0 434. log���TT
L
S
ECC
� 1 4 1
2
. log
d I d� 166 67
125. log ( )
B
C
