Mathematics for Computing. Lecture 2: Logarithms and indices

1. Mathematics for Computing

Lecture 2:
Logarithms and indices
Dr Andrew Purkiss
The Francis Crick Institute
or
Dr Oded Lachish, Birkbeck College
E-mail: [email protected]

2. Material

What are Logarithms?
Laws of indices
Logarithmic identities

3. Exponents

0
2 =1
1
2 = 2
2
2 = 2 x 2 = 4
23 = 2 x 2 x 2 = 8,
…
n
2 = 2 x 2 x … with n 2s
-1
2 ==
-2
2 = =
2-3 = =
…
2-n = =

4. Problem

We want to know how many bits the
number 456 will require when stored in
(non signed) binary format.
Solution based on what we learned last
week: Convert the number to Binary and
count the number of bits
After counting we get 9 (check it out)
There is a simpler way
Digit
numb
er
Numb
er
Remainder
when
dividing by 2
1
2
3
4
5
6
7
8
9
456
228
114
57
28
14
7
3
1
0
0
0
1
0
1
1
1
1

5. A simpler way

Round 456 up to the smallest power of 2 that is greater than 456.
Specifically, 512.
The answer!
9
Notice that 512 = 2 .
Why did we round up?
index
9
8
7
6
5
4
3
2
1
0
1
1
0
1
0
1
0
1
0
0
0
1
0
0
0
0
0
0
0
456
512
29
This gives us 2 to the power of the 1 + the index of the MSB of our number,
which is 1 less than its number of bits because the indices start from 0!

6. A simpler way

Much better, but we really don’t like the rounding up to the
smallest …
Don’t worry we just did this specific rounding up so that the
answer comes out nicely.
We will show a simpler way to do this (although we will start with
512 since it is nicer)

7. Logarithms

If we already knew the 512, then we would wonder which number is such
that
2x = 512
In words, how many times do we need to multiply 2 by itself to get 512?
The formal way to write this is x = log2512 , which means how many times do
we need to multiply 2 by itself to get 512?
We already know the answer is 9.
This is interpreted as follows:

8. Logarithms

We only know 456, lets compute log base 2 of 456
log2456 = 8.861…
Rounding this number up gives the answer we wanted, 9!
Why didn’t we get an integer? Because 456 is not a power of 2 so to get 456
we need to multiply 2 by itself 8.861 times, which can be done once we know
what this means.
So, how many bits do need in order to store the number 3452345 in binary
format?

9. Logarithms

If x = yz
then z = logy x

10. Logarithms and Exponents

If x = yz
then z = logy x
e.g. 1000 = 10 3,
then 3 = log10 (1000)
The base

11. Logarithms and Exponents: general form

From lecture 1) base index form:
number = baseindex
then index = logbase (number)

12. Graphs of exponents

x^2
x^3
10
30
9
20
8
7
10
6
5
X^2
X^3
0
-4
4
-3
-2
-1
0
-10
3
2
-20
1
-4
-3
-2
-1
0
0
x
1
2
3
4
-30
x
1
2
3
4

13. Graphs of logarithms

Log10
Log2
4
2
0
Logn(x)
0
2
4
6
-2
-4
-6
-8
x
8
10
12

14. Log plot

10^x
10^x
100000
100000
80000
10000
60000
1000
10^x
-5
10^x
-4
-3
-2
-1
40000
100
20000
10
0
0
x
1
2
3
4
5
-5
-4
-3
-2
-1
1
0
x
1
2
3
4
5

15. Three ‘special’ types of logarithms

Common Logarithm: base 10
Common in science and engineering
Natural Logarithm: base e (≈2.718).
Common in mathematics and physics
Binary Logarithm: base 2
Common in computer science

16. Laws of indices

1) a0 = 1
2) a1 = a

17. Laws of indices

1) a0 = 1
2) a1 = a
Examples:
20 = 1
100 = 1

18. Laws of indices

1) a0 = 1
2) a1 = a
Examples:
21 = 2
101 = 10

19. Laws of indices

3) a-x = 1/ax

20. Laws of indices

3) a-x = 1/ax
Example:
3-2 = 1/32 = 1/27

21. Laws of indices

4) ax · ay = a(x + y)
(a multiplied by itself x times) · (a multiplied by itself y times) = a multiplied by itself x+y times
5) ax / ay = a(x - y)
(a multiplied by itself x times) divided by (a multiplied by itself y times) = a multiplied by itself x-y times

22. Laws of indices

4) ax · ay = a(x + y)
42 · 43 = 4(2+3) = 45
16x64 = 1024
9 · 27 = 32 · 33 = 3(3 + 2) = 35= 243
25 · (1/5) = 52 · 5-1 = 5(2-1) = 51= 5

23. Laws of indices

5) ax / ay = a(x - y)
105 / 103 = 10(5-3) = 102
100,000 / 1,000 = 100
23 / 27 = 2(3-7) = 2-4
8 / 128 = 1/16, [24 = 16, 2-4 = 1/16, see law 3)]
64 / 4 = 26 / 22 = 2(6- 2) = 24 = 16
27 / 243 = 33 / 35 = 3(3 - 5) = 3-2= 1/9
25 / (1/5) = 52 / 5-1 = 5(2+1) = 53= 125

24. Laws of indices

6)
(ax)y = axy
(a multiplied by itself x times) multiplied by itself y times) = a multiplied by itself x ·y times
X times
X times
X times
(a ·a ·…) ·(a ·a ·…) ·…(a ·a ·…)
y times
7)
ax/y =
y
a1/y is the number you need to multiply by itself y times to get a. (a1/y)y = ay/y = a1 =a
1/2
1/3
So , 2 is square root of 2, which is, and 3 is square root of 3, which is,

25. Laws of indices

6) (ax)y = axy
(103)2 = 10(3x2) = 106
1,0002 = 1,000,000
(24)2 = 2(2x4) = 28
162 = 28 = 256
81 = (9) 2 = (32)2 = 34 = 81
1/16 = (1/4) 2 = (2-2)2 = 2-4 = 1/16

26. Laws of indices

7) ax/y = y√ax
10(4/2) = 2√104
102 = 2√10,000 = 100
2(9/3) = 3√29
23 = 3√512 = 8
8 = 23 = 26/2 = 2√64 = 8
1/7 = (7) -1 = (7) -2/2 = 2√(1/49) = 7

27. Logarithmic identities

‘Trivial’
Log form
logb 1 = 0
logb b = 1
Index form
b0 = 1
b1 = b

28. Logarithmic identities 2

y · logb x = logb xy
Definition of log
(bx)y = bxy
Definition of log

29. Logarithmic identities 2 examples

y · logb x = logb xy
(bx)y = bxy
Examples:
9 = 3 · log 8 = log 83 = log 512 = 9
2
2
2
512= (8)3 = (23)3 = 23·3= 29 = 512

30. Logarithmic identities 3

Negative Identity
-log x = log (1/x)
b
b
b-x = 1/bx
Addition
log x + log y = log xy
b
b
b
bx · by = b(x + y)
Subtraction
log x - log y = log x/y
b
b
b
bx / by = b(x - y)

31. Negative Identity

(3rd law of indices)
(definition of log)
Taking log from both
sides of the equation
(definition of log)

32. Negative identity

Negative Identity
-log x = log (1/x)
b
b
b-x = 1/bx
Examples:
-3 = -log 8 = log (1/8) = -3
2
2
1/8 = 2-3 = 1/23 =1/8

33. Addition identity

bx · by = b(x + y) (4th law of indices)
Taking log from
both sides of the
equation
Definition of log
(definition of log)

34. Addition identity examples

Addition
log x + log y = log xy
b
b
b
bx · by = b(x + y)
Examples:
5= 2+3 = log 4 + log 8 = log 4·8 = log 32 = 5
2
2
2
2
32= 4 · 8 = 22 · 23 = 2(2 + 3) = 25 = 32

35. Subtraction Identity

b ·b =b
x
y
(x + y)
(4 law of indices)
th
Taking log from
both sides of the
equation
Definition of log
(definition of log)
(definition of log + 3rd law of
indices )

36. Subtraction identity examples

Subtraction
log x - log y = log x/y
b
b
b
bx / by = b(x - y)
Examples:
-1 = 2-3 = log 4 - log 8 = log 4/8 = log 1/2 = -1
2
2
2
2
1/2= 4 / 8 = 22 / 23 = 2(2 - 3) = 2-1 = 1/2
3 = 5-2 = log2 32 - log2 4 = log2 32/4 = log2 8 = 3
8= 32 / 4 = 25 / 22 = 2(5 - 2) = 23 = 8

37. Changing the base

logb x = logy x / logy b
leads to
logb x = 1/(logx b)

38. Changing the base, examples 1

logb x = logy x / logy b
Examples:
2 = log4 16 = log2 16 / log2 4 = 4/2= 2
4 = log3 81 = log5 81 / log5 3

39. Changing the base, examples 2

logb x = 1/(logx b)
Examples:
2 = log 16 = 1/log 4 = 1/(1/2)= 2
4
16
4 = log3 81 = 1/ log81 3
= 1/(1/4)= 4