Презентация Mathematics for Computing. Lecture 2: Logarithms and indices онлайн
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Слайды и текст к этой презентации:
№1 слайд
Содержание слайда: Mathematics for Computing
№2 слайд
Содержание слайда: Material
What are Logarithms?
Laws of indices
Logarithmic identities
№3 слайд
Содержание слайда: Exponents
20 = 1
21 = 2
22 = 2 x 2 = 4
23 = 2 x 2 x 2 = 8,
…
2n = 2 x 2 x … with n 2s
№4 слайд
Содержание слайда: 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
№5 слайд
Содержание слайда: A simpler way
Round 456 up to the smallest power of 2 that is greater than 456.
Specifically, 512.
Notice that 512 = 29.
Why did we round up?
№6 слайд
Содержание слайда: 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 слайд
Содержание слайда: 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 слайд
Содержание слайда: 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 слайд
Содержание слайда: Logarithms
If x = yz
then z = logy x
№10 слайд
Содержание слайда: Logarithms and Exponents
If x = yz
then z = logy x
e.g. 1000 = 103,
then 3 = log10 (1000)
№11 слайд
Содержание слайда: Logarithms and Exponents: general form
From lecture 1) base index form:
number = baseindex
then index = logbase (number)
№12 слайд
Содержание слайда: Graphs of exponents
№13 слайд
Содержание слайда: Graphs of logarithms
№14 слайд
Содержание слайда: Log plot
№15 слайд
Содержание слайда: 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 слайд
Содержание слайда: Laws of indices
1) a0 = 1
2) a1 = a
№17 слайд
Содержание слайда: Laws of indices
1) a0 = 1
2) a1 = a
Examples:
20 = 1
100 = 1
№18 слайд
Содержание слайда: Laws of indices
1) a0 = 1
2) a1 = a
Examples:
21 = 2
101 = 10
№19 слайд
Содержание слайда: Laws of indices
3) a-x = 1/ax
№20 слайд
Содержание слайда: Laws of indices
3) a-x = 1/ax
Example:
3-2 = 1/32 = 1/27
№21 слайд
Содержание слайда: 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 слайд
Содержание слайда: 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 слайд
Содержание слайда: 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 слайд
Содержание слайда: 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
(a ·a ·…) ·(a ·a ·…) ·…(a ·a ·…)
7) ax/y =
a1/y is the number you need to multiply by itself y times to get a. (a1/y)y = ay/y = a1 =a
So , 21/2 is square root of 2, which is, and 31/3 is square root of 3, which is,
№25 слайд
Содержание слайда: 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 слайд
Содержание слайда: 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 слайд
Содержание слайда: Logarithmic identities
‘Trivial’
Log form Index form
logb 1 = 0 b0 = 1
logb b = 1 b1 = b
№28 слайд
Содержание слайда: Logarithmic identities 2
y · logb x = logb xy (bx)y = bxy
№29 слайд
Содержание слайда: Logarithmic identities 2 examples
y · logb x = logb xy (bx)y = bxy
Examples:
9 = 3 · log2 8 = log2 83 = log2 512 = 9
512= (8)3 = (23)3 = 23·3= 29 = 512
№30 слайд
Содержание слайда: Logarithmic identities 3
Negative Identity
-logb x = logb (1/x) b-x = 1/bx
Addition
logb x + logb y = logb xy bx · by = b(x + y)
Subtraction
logb x - logb y = logb x/y bx / by = b(x - y)
№31 слайд
Содержание слайда: Negative Identity
№32 слайд
Содержание слайда: Negative identity
Negative Identity
-logb x = logb (1/x) b-x = 1/bx
Examples:
-3 = -log2 8 = log2 (1/8) = -3 1/8 = 2-3 = 1/23 =1/8
№33 слайд
Содержание слайда: Addition identity
№34 слайд
Содержание слайда: Addition identity examples
Addition
logb x + logb y = logb xy bx · by = b(x + y)
Examples:
5= 2+3 = log2 4 + log2 8 = log2 4·8 = log2 32 = 5
32= 4 · 8 = 22 · 23 = 2(2 + 3) = 25 = 32
№35 слайд
Содержание слайда: Subtraction Identity
№36 слайд
Содержание слайда: Subtraction identity examples
Subtraction
logb x - logb y = logb x/y bx / by = b(x - y)
Examples:
-1 = 2-3 = log2 4 - log2 8 = log2 4/8 = log2 1/2 = -1
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 слайд
Содержание слайда: Changing the base
logb x = logy x / logy b
leads to
logb x = 1/(logx b)
№38 слайд
Содержание слайда: 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 слайд
Содержание слайда: Changing the base, examples 2
logb x = 1/(logx b)
Examples:
2 = log4 16 = 1/log16 4 = 1/(1/2)= 2
4 = log3 81 = 1/ log81 3 = 1/(1/4)= 4
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