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Advanced Mathematics Training Class Notes
Chapter 7: Logarithm and Number System
Chapter 7
Properties of Logarithm
Logarithm
Recall that:
1. am+n = am an
and Number System
2.
3.
4.
5.
Logarithm (對數
對數)
對數
Logarithm was originally invented for an easier multiplication and division. The use of
logarithm is to change multiplication and division to addition and subtraction. How to do so?
2
21
4
22
8
23
16
24
32
25
64
26
128
27
am
an
a = 1 (for a ≠ 0)
a1 = a
anc = (ac)n
0
By these, we can immediately see that:
1. loga m + loga n = loga (m n)
Take an example. It shouldn’t be hard to know 8 × 16 = 128. But doesn’t it also mean 23 × 24
= 27, or simply, “3 + 4 = 7”? Here we changed 8 into 3, 16 into 4. We calculated 3 + 4 = 7,
and change 7 back to 128. In the process of changing 8 into 3, etc, we are taking logarithm.
1
20
am−n =
256
28
2.
log a m − log a n = log a
3.
4.
5.
loga 1 = 0
loga a = 1
loga cn = n loga c
( )
m
n
(Addition property)
(Subtraction property)
(Logarithm of 1)
(Logarithm of base)
(Index property)
Moreover,
y
In general, if N can be written as x , then y is called the logarithm of N base x. It is written as
y = logx N.
For example, 3 = log2 8; -1 = log10 0.1.
If the base x is 10, we call this as common logarithm (常用對數), and denoted just by “log”.
For example, log 100 = 2.
6.
log a b =
log c b
log c a
(Base property)
7.
log a b =
1
logb a
(Inverse base property)
8.
log a b = log a n b n
9.
If the base is 2, it is sometimes denoted as “lg”. For example, lg 64 = 6.
c
log a a = a
b
log a c
(Base index property)
=c
(Cancellation property)
b log n a
10. a = n
(Index to base property)
These form the logarithm properties, which is useful in solving equations involving
logarithms and indices.
Besides these properties, you should also memorize the following values:
log 2 = 0.301, log 3 = 0.477
Finally, if a number < 0 is taken logarithm (logn x, x < 0), the result is undefined.
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Advanced Mathematics Training Class Notes
Chapter 7: Logarithm and Number System
Base-n Number System
Converting Number between Binary and Decimal
In daily life, we express numbers in decimal system (十進制), but sometimes it would be
more useful in using other bases, like binary system (二進制) in computer science.
Converting binary number to decimal number
It is easy. Take an example, to convert 10111010012 to decimal:
Label the numbers with 0, 1, 2, etc. from right hand side at the unit (個位):
9876543210
1011101001
A base-n number system is that, when you count numbers from 1, 2, 3, etc., you need to add
one more digit when reaching n, and reset the current digit to 0.
For example, in decimal (base-10):
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, …
Remove the column having “0” as the digit:
976530
111111
In binary (base-2):
1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, …
In ternary (base-3):
1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110, 111, 112, 120, 121, 122, 200, …
Remove the last row: 9,7,6,5,3,0
Raise power of 2 from them: 512,128,64,32,8,1
Add them altogether: 745. This is what 10111010012 represents.
If the base is higher than 10, like 16, the digits (0~9) are not enough. We will use alphabets
(A~Z) in this case. For example, in hexadecimal (base-16):
Another example: 1100.101012.
3 2 1 0 -1 -2 -3 -4 -5
11001 0 1 0 1
1, 2, 3, …, 8, 9, A, B, C, D, E, F, 10, 11, …, 19, 1A, 1B, 1C, 1D, 1E, 1F, 20, …
Base
2
3
4
5
8
10
16
3 2 -1 -3 -5
111 1 1
Name
Binary (or “Bin”)
Ternary
Quaternary
Quinary
Octal (or “Oct”)
Decimal (or “Dec”)
Hexadecimal (or “Hex”)
23 + 22 + 2-1 + 2-3 + 2-5 = 12.6562510
Converting decimal number to binary number
We use short division to do this. For example, change 12310 to binary.
Divide 123 by 2, write down the remainder, and repeat until zero is reached:
A base-n number is usually written end with “n”, like 1010112.
A
base-n
number
( a j …a2a1a0 )n
can
be
considered
as
a j n j + a j −1n j −1 + … + a2n 2 + a1n + a0 .
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Advanced Mathematics Training Class Notes
Chapter 7: Logarithm and Number System
2 123 …1
2 61 …1
2 30 … 0
2 15 …1
2 7 …1
2 3 …1
2 1 …1
0
Read from bottom to top, 1111011. This is the representation of 12310 in binary.
Revision
For decimal numbers with fractional part (小數部), we separate the fractional part from its
integral part (整數部). Like 7.687510, the integral part is 7, and the fractional part is 0.6875.
Convert the integral part into binary directly (1112), and do the fractional part as follows:
Exercise
0.6875 ×2 = 1.375 …1
0.375 ×2 = 0.75 … 0
0.75 ×2 = 1.5
…1
×2 = 1
0.5
…1
0
We multiply 2 to the original fractional part. If the resulting integral part is 1, take away that,
and repeat the process with the remaining fractional part until reaching 0. Read from top,
1011. This is the fractional part in binary. On the whole, 7.687510 = 111.10112.
1. Given log 2 = a, log 3 = b. Express the followings in terms of a and b:
a) log 5
b) log 9
c) log 270
d) lg 30
e) log5 6
2. Solve the following equations:
a) log(2x – 4) = 3
b) log(x – 1) + log(x + 3) = log 5
c) log x – log (3x2 – 5) = -1
d) (log x)2 – log x2 = 3
e) 52x – 6(5x+1) + 125 = 0
3. Which is bigger: 275, 350, 525? Give reasons.
4. (HKMO 2004 Heat) Find log1400 2 + log1400 3 5 + log1400 6 7 .
In this chapter, we’ve learnt:
1. What is logarithm
2. Properties of logarithm
3. Base-n number system
4. Converting number between binary and decimal
5. Application of base-n numbers in solving equations involving indices
In the followings, if not specified, x is the variable, and k is a positive number.
For the other bases, the converting process is similar, and these are left to the readers to
examine.
Application of Base-n Numbers in Solving Equations
Involving Indices
1 1
+ .
a b
w
6. (ISMC 2000 Final) Find integral solutions of 2 + 2x + 2y + 2z = 20.625, where w>x>y>z.
7. (ISMC 2000 Final) Find a sequence a that is composed of 14 positive integers such that:
5. (HKMO 2001 Heat) If 4a = 25b = 10, find out
Base-n numbers other useful in solving equations like na + nb + nc + … + nz = X. It would be
very easy if you convert X into base-n, which is nearly impossible using logarithm only.
w
x
y
14
∑ 3a
k
k =1
z
For example, solve 2 + 2 + 2 + 2 = 42.5.
Convert 42.5 into binary (101010.12), it is then clear that w = 5, x = 3, y = 1, z = -1.
43
= 6558
Advanced Mathematics Training Class Notes
Chapter 7: Logarithm and Number System
8. (ISMC 2000 Final) Given:
(
x = x + log ( x
x2 = x1 + log x12 + x12 + 1
3
2
2
2
)
+ x22 + 1
)
⋮
(
x1 = x2000 + log x20002 + x20002 + 1
)
Find x2000. [Hint: Property #3]
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Advanced Mathematics Training Class Notes
Chapter 7: Logarithm and Number System
Suggested Solutions for the Exercise
1a) 1 – a
b) 2b
c) 3b + 1
d) 1+ab
e)
b+a
1− a
2a) 502
b) –4, 2
5 ± 2 10
c)
3
d) 1/10, 1000
e) 1, 2
3) 350
4) 1/6
5) 2
6) 4, 2, -1, -3
7) 1,1,2,2,3,3,4,4,5,5,6,6,7,7
8) 0
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