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Page 1: Mathematics Number:  Logarithms

MathematicsNumber: Logarithms

Science and Mathematics Education Research Group

Supported by UBC Teaching and Learning Enhancement Fund 2012-2014

Department of Curriculum and Pedagogy

FACULTY OF EDUCATIONa place of mind

Page 2: Mathematics Number:  Logarithms

Logarithms

Page 3: Mathematics Number:  Logarithms

Introduction to Logarithms

Find the value of x where 2x = 8.

16E.

8D.

4C.

3B.

2A.

x

x

x

x

x

Press for hint

Write 8 as a power with base 2.

Exponent law:

yxaa yx ifonly and if

Page 4: Mathematics Number:  Logarithms

Solution

Answer: B

Justification: Try writing the right hand side of the equation as a power of two.

Then use the exponent law: yxaa yx ifonly and if

3

22

823

x

x

x

328

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Introduction to Logarithms II

Find the value of x where 2x = 18.

above theof NoneE.

6D.

5C.

4B.

3A.

x

x

x

x

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Solution

Answer: E

Justification: We cannot write 18 as a power of 2. This means we cannot solve this question the same way as the previous one.

The answer cannot be written as an integer, so the correct answer is E, “None of the above.”

In order to answer this question, we will have to learn about logarithms.

x 2x

3 8

4 16

5 32

6 64

Although we can’t find the exact answer, we can guess the answer is between 4 and 5.

Page 7: Mathematics Number:  Logarithms

Introduction to Logarithms III

If y = ax then logay = x. Notice how the logarithm (logay) returns

the exponent of ax.

Using the above definition, what is the value of log28?

above theof NoneE.

58logD.

48logC.

38logB.

28logA.

2

2

2

2

Page 8: Mathematics Number:  Logarithms

Solution

Answer: B

Justification: This question is exactly the same as question 1. We must find the value of x where 2x = 8.

In question 1, we were asked to find the exponent x such that 2x = 8. This is exactly the same as finding the value of log28.

In general, we can convert between logs and exponents as follows:

82since38log 32

loga y = x

y = axA helpful reminder:

loga y = xa to the x

equals y

Page 9: Mathematics Number:  Logarithms

Introduction to Logarithms IV

above theof NoneE.

logD.

logC.

2logB.

18logA.

2

2

18

2

yx

xx

x

x

Find the value of x where 2x = 18. Express the answer using logarithms.

Hint: Try expressing the question in terms of logarithms.

Press for hint

If y = ax then logay = x.

loga y = x

y = ax

Page 10: Mathematics Number:  Logarithms

Answer: A

Justification: Recall how equations in the form y = ax, can be rewritten with logarithms, and vice-versa:

Notice how applying this to 2x = 18 allows us to solve for x, which we were not able to do without logarithms:

This is the exact answer to the equation above, so . We will see later that this result can be generalized to a property of logarithms:

Solution

18log

182

2

x

x

182 18log2

ab ab log

loga y = x

y = ax

Page 11: Mathematics Number:  Logarithms

Logarithm Laws

The following is a summary of important logarithm properties:

baab logloglog

bab

alogloglog

anan loglog

b

aa

c

cb log

loglog

01log b

Other useful properties include:

1log bb ab ab log

aa bb log1

log

yxyx bb ifonly and ifloglog

Note: When you see log x, it is assumed that the base of the logarithm is 10.

xx 10loglog

When you see ln x, the base is assumed

to be e ≈ 2.72.

xx elogln

Page 12: Mathematics Number:  Logarithms

Properties of Logarithms I

What is written as a single logarithm?

ab

c

c

ab

c

ab

abc

cba

7

7

7

7

7

7

logE.

logD.

log

logC.

logB.

logA.

cba 777 logloglog

Page 13: Mathematics Number:  Logarithms

Solution

Answer: D

Justification: We will use the two following two properties to simplify the expression:

First combine the terms using the first property 1 shown above:

Next combine the term that is subtracted using property 2:

baab logloglog.1 bab

alogloglog.2

cabcba 77777 logloglogloglog

abba 777 logloglog

c

abcab 777 logloglog

Does order matter? Would simplifying the subtracted term first change the answer?

Page 14: Mathematics Number:  Logarithms

Properties of Logarithms II

What is the value of log101008.

8E.

16D.

10C.

10B.

100A.

8

16

8

Page 15: Mathematics Number:  Logarithms

Solution

Answer: D

Justification: Use exponent and logarithm laws to simplify the problem. We can start by pulling the exponent the outside of the logarithm:

The solution tells us that 1016 = 1008.

210log100logsince

loglogsince2

1010

ana bn

b

16

28

100log8100log 108

10

Page 16: Mathematics Number:  Logarithms

Alternative Solution

Answer: D

Justification: Since the logarithm is in base 10, another strategy is to write all terms with a base of 10.

1logsince

loglogsince

)(since

b

ana

aa

b

bn

b

xyyx

16

10log16

10log

10log100log

10

1610

8210

810

Page 17: Mathematics Number:  Logarithms

Properties of Logarithms III

Which of the following is not equivalent to the following equation, where b > 0, b ≠ 1?

cb

ba

cabb

ca

cabb

bab

x

bx

b

bb

bb

bc

log

logE.

logD.

1logC.

loglogB.

A.

cbabb log

Page 18: Mathematics Number:  Logarithms

Solution

Answer: D

Justification: Answer A writes the logarithm in exponential form:

We can also use the property log xy = log x + log y to break up the log:

When we let , we get answer C.

When we let , we get answer B.

bcbb babcba log

bbb

bb abba logloglog

1 ab blog

1log bb

aba bb

b loglog

Answer continues on the next slide

Page 19: Mathematics Number:  Logarithms

Solution Continued

Answer E uses the change of base property to rewrite the logarithm with base x:

Answer D is very similar to answers B and C, although . Therefore answer D is the only non-equivalent expression:

b

baba

x

bxb

b log

loglog

bbb log

abbba bb

b loglog

Page 20: Mathematics Number:  Logarithms

Properties of Logarithms IV

Which of the following is equivalent to the expression shown:

a2E.

logD.

2logC.

log2logB.

2logA.

2

1

b

b

bb

b

ab

ab

ka

k

ak

b

b

k

ak

log

2log

Press for hint

y

xx

c

cy log

loglog

Change of base property:

Page 21: Mathematics Number:  Logarithms

Solution

Answer: C

Justification: The most common mistake that may arise when answering this question is incorrectly stating:

Making this error will lead to answer A or B.

Instead, we can use the change of base property to write the expression as a single logarithm:

b

b

b

b a

kk

ak 2

loglog

2log

ab

k

ak b

b

b2log

log

2log

y

xx

c

cy log

loglog

Change of base property:

This statement is not correct

Page 22: Mathematics Number:  Logarithms

Properties of Logarithms V

Which of the following are equivalent to the following expression

above theof NoneE.

)log(D.

)log(C.

loglogB.

loglogA.

nn

n

ba

ban

bnan

ba

nba )log(

Page 23: Mathematics Number:  Logarithms

Solution

Answer: E

Justification: Expressions in the form and cannot be simplified. Therefore, cannot be simplified.

The two most common errors are:

Compare the incorrect statements above with the correct logarithm laws:

baba loglog)log( ana n loglog

baab loglog)log( )log()log( anan

)log( ba na)log( nba )log(

Page 24: Mathematics Number:  Logarithms

Properties of Logarithms VI

If a > b > 0, how do log(a) and log(b) compare?

ba

ba

ba

ba

ba

loglogE.

loglogD.

loglogC.

loglogB.

loglogA.

Page 25: Mathematics Number:  Logarithms

Solution

Answer: A

Justification: It may be easier to compare logarithms by writing them as exponents instead. Let:

Converting to exponents gives: and

Aa log Bb log

aA 10 bB 10

Substitute a = 10A, b = 10B

The exponent A must be larger than the exponent B

Substitute A = log a, B = log bba

BA

loglog

BA

ba

1010

Given in the question

The larger the exponent on the base 10, the larger our final value. This is true whenever the bases are the same and greater than 1. For example, 104 > 103.

Page 26: Mathematics Number:  Logarithms

Properties of Logarithms VII

What is the value of the following expression?

simplified beCannot E.

D.

C.

B.

A.

b

a

a

b

b

a

abblog

Page 27: Mathematics Number:  Logarithms

Solution

Answer: A

Justification: This is one of the logarithm laws:

If logba = x, x is the exponent to which b must be raised in order to

equal a (in order words, bx = a) by the definition of logarithms.

It is important to remember that logarithms return the value of an exponent. When the exponent returned is the original logarithm, we get the property:

ab ab log

logb a= logbab to the logba

equals a

logb a = xb to the x

equals a

ab ab log

Page 28: Mathematics Number:  Logarithms

Alternative Solution

Answer: A

Justification: We can also solve this problem using other properties of logarithms. Let . The value of x will be our answer. If we write this equation using logarithms:

We can now use the property logbx = logba if and only if x = a.

ax

xb

bb

ab

loglog

log

xb ab log

logb x = y

x = by In this question, ay blog

ab

axab

log

Page 29: Mathematics Number:  Logarithms

Properties of Logarithms VIII

If a > b > 1 and c > 1 how do loga(c) and logb(c) compare?

)(log)(logE.

)(log)(logD.

)(log)(logC.

)(log)(logB.

)(log)(logA.

cc

cc

cc

cc

cc

ba

ba

ba

ba

ba

Press for hint

Use the change of base property:

)log(

)log()(log

a

cca

)log(

)log(log

b

ccb

Page 30: Mathematics Number:  Logarithms

Solution

Answer: C

Justification: We can use the change of base property to make the two expressions easier to compare:

Since a, b, and c are greater than 1, log(a), log(b), and log(c) are all positive. From the previous question we learned that log(a) > log(b) if a > b. Since log(a) is in the denominator, it will produce a smaller fraction than log(b).

)log(

)log()(log

a

cca

)log(

)log(log

b

ccb

)log(

)log(

)log(

)log(

b

c

a

c )(log)(log cc ba

Page 31: Mathematics Number:  Logarithms

Alternative Solution

Answer: C

Justification: You can also change the question to exponents and compare. Let:

If we equate these two equations we get:

Since a > b, the exponent A must be smaller than B. If this were not the case, the LHS will have a larger base and exponent than the RHS, so they cannot be equal.

ca

AcA

a

)(log

cb

BcB

b

)(log

BA ba

)(log)(log ccBA ba

Page 32: Mathematics Number:  Logarithms

Properties of Logarithms IX

Given log37 ≈ 1.7712, what is the approximate value of log349 ?

9408.15)7712.1(3E.

3984.12)7712.1(7D.

5424.3)7712.1(2C.

13.3)7712.1(B.

3.17712.1A.

2

2

Press for hint

Write log349 in terms of log37 using the following property:

anan loglog

Page 33: Mathematics Number:  Logarithms

Solution

Answer: C

Justification: Without knowing that log37 ≈ 1.7712, we can still determine an approximate value for log349. Since log327 = 3 and log381 = 4, we should expect log349 is between 3 and 4. We can already rule out answers A, D, and E.

To get a better estimate, we can use the properties of logarithms:

5424.3

)7712.1(2

7log2

7log49log

3

233

anan loglog 7712.17log3

since

since

Page 34: Mathematics Number:  Logarithms

Properties of Logarithms X

If what does equal?

100E.

10D.

1C.

1.0B.

0A.

1log

log

10

10 n

m

n

m100log

Press for hint

Write as1log

log

10

10 n

mnm 1010 loglog

Page 35: Mathematics Number:  Logarithms

Solution

Answer: A

Justification: If we simplify the equation , we can find the relationship between m and n:

We can now plug this value into the other expression:

1

loglog

1log

log

1010

10

10

n

m

nm

nm

n

m

01loglog 100100

n

msince 1

n

m

1log

log

10

10 n

m

Page 36: Mathematics Number:  Logarithms

Alternative Solution

Answer: A

Justification: The solution can also be found by simplifying

first, although much more work is required to get a final answer:

02

log

2

log2

log

2

log

100log

log

100log

log

logloglog

1010

1010

10

10

10

10

100100100

mm

nm

nm

nmn

m

Change to base 10

Since log10m = log10n

log10100 = 2

n

m100log