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Mathematics for Computing

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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

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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

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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?

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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)

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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:

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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?

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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)

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

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Graphs of exponents

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Graphs of logarithms

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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

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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

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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

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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

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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

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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

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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,

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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

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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

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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

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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

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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)

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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

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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

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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