Mathematics Number: Logarithms - University of British ...

36
Mathematics Number: Logarithms Science and Mathematics Education Research Group Supported by UBC Teaching and Learning Enhancement Fund 2012-2014 Department of Curriculum and Pedagogy FACULTY OF EDUCATION a place of mind

Transcript of Mathematics Number: Logarithms - University of British ...

Page 1: Mathematics Number: Logarithms - University of British ...

Mathematics

Number: Logarithms

Science and Mathematics

Education Research Group

Supported by UBC Teaching and Learning Enhancement Fund 2012-2014

Department of

Curr iculum and Pedagogy

F A C U L T Y O F E D U C A T I O N a place of mind

Page 2: Mathematics Number: Logarithms - University of British ...

Logarithms

Page 3: Mathematics Number: Logarithms - University of British ...

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 - University of British ...

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

82

3

x

x

x

328

Page 5: Mathematics Number: Logarithms - University of British ...

Introduction to Logarithms II

Find the value of x where 2x = 18.

above theof NoneE.

6D.

5C.

4B.

3A.

x

x

x

x

Page 6: Mathematics Number: Logarithms - University of British ...

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 - University of British ...

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 - University of British ...

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 3

2

loga y = x

y = ax A helpful reminder:

loga y = x

a to the x

equals y

Page 9: Mathematics Number: Logarithms - University of British ...

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 - University of British ...

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

18218log2

abab

log

loga y = x

y = ax

Page 11: Mathematics Number: Logarithms - University of British ...

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 abab

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 - University of British ...

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 - University of British ...

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 - University of British ...

Properties of Logarithms II

What is the value of log101008.

8E.

16D.

10C.

10B.

100A.

8

16

8

Page 15: Mathematics Number: Logarithms - University of British ...

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

loglogsince

2

1010

ana b

n

b

16

28

100log8100log 10

8

10

Page 16: Mathematics Number: Logarithms - University of British ...

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

b

n

b

xyyx

16

10log16

10log

10log100log

10

16

10

82

10

8

10

Page 17: Mathematics Number: Logarithms - University of British ...

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

b

x

b

b

b

bb

bc

log

logE.

logD.

1logC.

loglogB.

A.

cbab

b log

Page 18: Mathematics Number: Logarithms - University of British ...

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.

bcb

b babcba log

b

bb

b

b abba logloglog

1 ab blog

1log bb

aba b

b

b loglog

Answer continues on the next slide

Page 19: Mathematics Number: Logarithms - University of British ...

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

b

xb

blog

loglog

bbb log

abbba b

b

b loglog

Page 20: Mathematics Number: Logarithms - University of British ...

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

a

b

k

a

k

a

k

b

b

k

a

k

log

2log

Press for hint

y

xx

c

cy

log

loglog

Change of base property:

Page 21: Mathematics Number: Logarithms - University of British ...

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

k

k

a

k 2log

log

2log

a

b

k

a

k bb

b2log

log

2log

y

xx

c

cy

log

loglog

Change of base property:

This statement is not correct

Page 22: Mathematics Number: Logarithms - University of British ...

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 - University of British ...

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( anan

loglog

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

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

Page 24: Mathematics Number: Logarithms - University of British ...

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 - University of British ...

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 b ba

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 - University of British ...

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 - University of British ...

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:

abab

log

logb a= logba

b to the logba

equals a

logb a = x

b to the x

equals a

abab

log

Page 28: Mathematics Number: Logarithms - University of British ...

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

xbab

log

logb x = y

x = by In this question,

ay blog

ab

ax

ab

log

Page 29: Mathematics Number: Logarithms - University of British ...

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 - University of British ...

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 - University of British ...

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

Ac

A

a

)(log

cb

Bc

B

b

)(log

BA ba

)(log)(log ccBA ba

Page 32: Mathematics Number: Logarithms - University of British ...

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 - University of British ...

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

2

33

anan loglog

7712.17log3

since

since

Page 34: Mathematics Number: Logarithms - University of British ...

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 as 1log

log

10

10 n

mnm 1010 loglog

Page 35: Mathematics Number: Logarithms - University of British ...

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 - University of British ...

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:

0

2

log

2

log

2

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