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
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
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
Introduction to Logarithms II
Find the value of x where 2x = 18.
above theof NoneE.
6D.
5C.
4B.
3A.
x
x
x
x
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.
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
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
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
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
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
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
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?
Properties of Logarithms II
What is the value of log101008.
8E.
16D.
10C.
10B.
100A.
8
16
8
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
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
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
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
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
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:
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
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(
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(
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.
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.
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
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
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
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
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
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
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
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
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
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
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
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