Algorithms and Discrete Mathematics 20082009

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Algorithms and Discrete Mathematics 2008/2009

• Lecture 5
• Exponentiation and logarithms II

Ioannis Ivrissimtzis
10-Nov-2008
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Overview of the lecture

• Logarithms
• Properties of logarithms
• Computation of square roots
• Computations of logarithms

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Logarithms

• For real positive numbers x,b with b?1, the
  logarithm of x to the base
• b, written logbx is the unique real number y that
  satisfies byx.
• That is, if we raise b to the power of logbx we
  get x.
• When it does not create confusion, we do not put
  argument of the
• logarithm in brackets. That is, we write logbx
  rather than logb(x).

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Logarithms

• Example 5.1

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Logarithms

• Example 5.2 Let b be a positive real number,
  b?1. We have

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Logarithms

• The logarithms are inverses
• of the exponentials.
• They are only defined on
• positive real numbers.
• For any base, the logarithm
• of 1 is 0.
• For bgt1 they increase
• monotonically. They grow
• slow.

1
0
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Overview of the lecture

• Logarithms
• Properties of logarithms
• Computation of square roots
• Computations of logarithms

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Properties of logarithms

• Proposition 5.1 Let b,r,s be positive real
  numbers with b?1. We have
• These properties are related to properties of the
  exponents mentioned
• in the previous lecture.

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Properties of logarithms

• Proof of the first property
• By the definition of logarithm
• Using a property of the exponents, we get
• and finally, again from the definition of the
  logarithms, we get

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Properties of logarithms

• Proposition 5.2 Let a,b,x be positive real
  numbers, with a,b ?1. We have,
• This proposition allows the computation of the
  logarithm to any base, as
• long as we can compute the logarithm to a given
  base.

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Properties of logarithms

• Proof By the definition of the logarithm
• Taking logarithms (to base b) on both sides
• Using proposition 5.1 (3) we get
• giving,

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Overview of the lecture

• Logarithms
• Properties of logarithms
• Computation of square roots
• Computations of logarithms

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Computation of square roots

• An iterative algorithm for computing the square
  root of a number b
• 1. Start with an initial guess of the square
  root x1
• 2. At each iteration of the algorithm, update x
  by
• 3. Stop when the algorithm converges, that is,
  there is small difference in the value of x
  between two iterations.

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Computation of square roots

• Pseudo-code for computing the square root of a
  positive number b

procedure square root ( b positive real number
) x 1 while ( the algorithm has not
converged yet ) x is the square root of b
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Computation of square roots

• This algorithm is iterative. It is not
  immediately clear why the value of x
• approaches the square root of b.
• The algorithm has a simple geometric
  interpretation.
• It was known (or something equivalent to it) to
  the Babylonians
• 2000BC ?.
• You dont have to memorize the algorithm.

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Computation of square roots
The YBC 7289 tablet photo by Bill Casselman Yale
Babylonian Collection

• Fowler and Robson, 1998
• reprinted from Aaboe 1964

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Overview of the lecture

• Logarithms
• Properties of logarithms
• Computation of square roots
• Computations of logarithms

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The natural logarithm

• The logarithm with base
• arises naturally in many applications.
• It is called the natural logarithm and denoted
  lnx.
• e is an important mathematical constant (like the
  p 3.1415 )

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Series expansion

• If we can compute natural logarithms, than using
  Proposition 5.2 we
• can compute logarithms at nay base.
• We can compute natural logarithms using the
  series expansion
• A series has infinite many terms.
• The terms of this series decrease in absolute
  value (because -1ltxlt1)
• and so, it is the first few terms that mainly
  influence the result.

for
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Series expansion

• We can find a good approximation of the logarithm
  by computing an
• initial part of the series.
• The larger the initial part (i.e. the most terms
  we take into account) the
• better the approximation.
• Different series can be used in the computation
  of other functions, for
• example the trigonometric functions.
• They can also be used to generalize a function
  beyond the realm of the
• real numbers, e.g. matrix logarithms.