🗊Презентация Mathematics for Computing. Lecture 2: Logarithms and indices

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Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №1Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №2Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №3Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №4Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №5Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №6Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №7Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №8Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №9Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №10Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №11Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №12Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №13Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №14Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №15Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №16Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №17Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №18Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №19Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №20Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №21Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №22Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №23Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №24Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №25Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №26Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №27Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №28Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №29Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №30Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №31Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №32Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №33Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №34Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №35Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №36Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №37Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №38Mathematics for Computing. Lecture 2: Logarithms and indices, слайд №39
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Mathematics for Computing
Описание слайда:
Mathematics for Computing

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Material What are Logarithms? Laws of indices Logarithmic identities
Описание слайда:
Material What are Logarithms? Laws of indices Logarithmic identities

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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
Описание 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

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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
Описание 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

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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?
Описание 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?

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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)
Описание 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)

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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:
Описание 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:

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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?
Описание 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?

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Logarithms If x = yz then z = logy x
Описание слайда:
Logarithms If x = yz then z = logy x

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Logarithms and Exponents If x = yz then z = logy x e.g. 1000 = 103, then 3 = log10 (1000)
Описание слайда:
Logarithms and Exponents If x = yz then z = logy x e.g. 1000 = 103, then 3 = log10 (1000)

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Logarithms and Exponents: general form From lecture 1) base index form: number = baseindex then index = logbase (number)
Описание слайда:
Logarithms and Exponents: general form From lecture 1) base index form: number = baseindex then index = logbase (number)

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Graphs of exponents
Описание слайда:
Graphs of exponents

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Graphs of logarithms
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Graphs of logarithms

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Log plot
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Log plot

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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
Описание 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

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Laws of indices 1) a0 = 1 2) a1 = a
Описание слайда:
Laws of indices 1) a0 = 1 2) a1 = a

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Laws of indices 1) a0 = 1 2) a1 = a Examples: 20 = 1 100 = 1
Описание слайда:
Laws of indices 1) a0 = 1 2) a1 = a Examples: 20 = 1 100 = 1

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Laws of indices 1) a0 = 1 2) a1 = a Examples: 21 = 2 101 = 10
Описание слайда:
Laws of indices 1) a0 = 1 2) a1 = a Examples: 21 = 2 101 = 10

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Laws of indices 3) a-x = 1/ax
Описание слайда:
Laws of indices 3) a-x = 1/ax

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Laws of indices 3) a-x = 1/ax Example: 3-2 = 1/32 = 1/27
Описание слайда:
Laws of indices 3) a-x = 1/ax Example: 3-2 = 1/32 = 1/27

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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
Описание 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

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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
Описание 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

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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
Описание 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

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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,
Описание 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,

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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
Описание 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

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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
Описание 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

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Logarithmic identities ‘Trivial’ Log form Index form logb 1 = 0 b0 = 1 logb b = 1 b1 = b
Описание слайда:
Logarithmic identities ‘Trivial’ Log form Index form logb 1 = 0 b0 = 1 logb b = 1 b1 = b

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Logarithmic identities 2 y · logb x = logb xy (bx)y = bxy
Описание слайда:
Logarithmic identities 2 y · logb x = logb xy (bx)y = bxy

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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
Описание 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

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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)
Описание 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)

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Negative Identity
Описание слайда:
Negative Identity

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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
Описание 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

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Addition identity
Описание слайда:
Addition identity

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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
Описание 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

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Subtraction Identity
Описание слайда:
Subtraction Identity

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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
Описание 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

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Changing the base logb x = logy x / logy b leads to logb x = 1/(logx b)
Описание слайда:
Changing the base logb x = logy x / logy b leads to logb x = 1/(logx b)

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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
Описание 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

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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
Описание 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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