Mathematics Primer Logarithm

1. Beginning Direct3D Game Programming:
Mathematics 2
Logarithm
jintaeks@gmail.com
Division of Digital Contents, DongSeo University.
March 2016

2. Exponentiation
 Exponentiation is
a mathematical operation,
written as bn, involving two
numbers, the base b and
the exponent n.
 When n is a positive integer,
exponentiation corresponds to
repeated multiplication of the
base: that is, bn is the product of
multiplying n bases:
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3. Logarithm
 In mathematics, the logarithm is the inverse
operation to exponentiation.
 That means the logarithm of a number is the exponent to
which another fixed value, the base, must be raised to
produce that number.
 In simple cases the logarithm counts repeated multiplication.
 For example, the base 10 logarithm of 1000 is 3, as 10 to the
power 3 is 1000 (1000 = 10 × 10 × 10 = 103); the
multiplication is repeated three times.
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4.  The logarithm of x to base b(log base b of x),
denoted logb(x), is the unique real number y such that
by = x.
 For example, as 64 = 26, we have
log2(64) = 6
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5.  The graph of the logarithm to base 2 crosses
the x axis (horizontal axis) at 1 and passes through the points
with coordinates (2, 1), (4, 2), and (8, 3). For example, log2(8) =
3, because 23 = 8. The graph gets arbitrarily close to
the y axis, but does not meet or intersect it.
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6.  A full 3-ary tree can be used to visualize the exponents of 3
and how the logarithm function relates to them.
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7. Product, quotient, power and root
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8. Binary search
 In computer science, binary search, also known as half-
interval search or logarithmic search,[1] is a search
algorithm that finds the position of a target value within
a sorted array.
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9. 9
: Number of iterations for binary
and linear search for n from 1 to
1000.

10. Practice
 Write a binary search program.
 Use data {1,3,5,7,9,10,15,16}.
 Print the number of comparison when function finished.
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11. Time complexity
 In computer science, the time complexity of
an algorithm quantifies the amount of time taken by an
algorithm to run as a function of the length of the string
representing the input.
 The time complexity of an algorithm is commonly expressed
using big O notation, which excludes coefficients and lower
order terms.
 When expressed this way, the time complexity is said to be
described asymptotically, i.e., as the input size goes to
infinity.
 For example, if the time required by an algorithm on all inputs
of size n is at most 5n3 + 3n for any n (bigger than some n0),
the asymptotic time complexity is O(n3).
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12. Sort Algorithms
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13. Quick sort
 Quicksort is a fast sorting algorithm, which is used not only
for educational purposes, but widely applied in practice.
 On the average, it has O(n log n) complexity, making
quicksort suitable for sorting big data volumes.
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14. Algorithm
 The divide-and-conquer strategy is used in quicksort. Below
the recursion step is described:
1. Choose a pivot value. We take the value of the middle
element as pivot value, but it can be any value, which is in range
of sorted values, even if it doesn't present in the array.
2. Partition. Rearrange elements in such a way, that all elements
which are lesser than the pivot go to the left part of the array
and all elements greater than the pivot, go to the right part of
the array. Values equal to the pivot can stay in any part of the
array. Notice, that array may be divided in non-equal parts.
3. Sort both parts. Apply quicksort algorithm recursively to the
left and the right parts.
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15. Partition algorithm in detail
 There are two indices i and j and at the very beginning of
the partition algorithm i points to the first element in the
array and j points to the last one.
 Then algorithm moves i forward, until an element with value
greater or equal to the pivot is found.
 Index j is moved backward, until an element with value lesser
or equal to the pivot is found.
 If i ≤ j then they are swapped and i steps to the next position
(i + 1), j steps to the previous one (j - 1).
 Algorithm stops, when i becomes greater than j.
 After partition, all values before i-th element are less or equal
than the pivot and all values after j-th element are greater or
equal to the pivot.
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16. example
 Sort {1, 12, 5, 26, 7, 14, 3, 7, 2} using quicksort.
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18. Practice
 Write a Quick sort function.
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19. Singularity
 In mathematics,
a singularity is in general a
point at which a given
mathematical object is not
defined, or a point of an
exceptional set where it fails
to be well-behaved in some
particular way, such
as differentiability.
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20.  The function g(x) = |x| also has a singularity at x= 0, since it is
not differentiable there.
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21.  The graph defined by y2 = x also has a singularity at (0,0), this
time because it has a "corner" (vertical tangent) at that point.
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22. References
 https://en.wikipedia.org/wiki/Logarithm
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