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# 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

02 =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 thenumber 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 thesmallest …

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 suchthat

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 456log2456 = 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 = yzthen z = logy x

## 10. Logarithms and Exponents

If x = yzthen 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^2x^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

Log10Log2

4

2

0

Logn(x)

0

2

4

6

-2

-4

-6

-8

x

8

10

12

## 14. Log plot

10^x10^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 10Common 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 = 12) a1 = a

## 17. Laws of indices

1) a0 = 12) a1 = a

Examples:

20 = 1

100 = 1

## 18. Laws of indices

1) a0 = 12) a1 = a

Examples:

21 = 2

101 = 10

## 19. Laws of indices

3) a-x = 1/ax## 20. Laws of indices

3) a-x = 1/axExample:

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√ax10(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 xyDefinition 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

Additionlog 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 =bx

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

Subtractionlog 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 bleads to

logb x = 1/(logx b)

## 38. Changing the base, examples 1

logb x = logy x / logy bExamples:

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