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    Exponent and Logarithm

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    Overview

    Exponent Logarithm

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    Problems in Real life

    How do I calculate real quantities which arenot linear but power of some quantity.

    How do I calculate the amount of time ittakes to decay a radioactive substance to

    half its original amount (calledHalf Life).

    How do I find the age of a fossil/rock.

    How do I amortize my loan.

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    Solution

    All these problems can be solved bymodeling these events as a function of some

    exponent.

    Let us learn some formal theories relating to

    these exponents.

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    Exponent A base in mathematics is used to refer to a

    particular mathematical object that is usedas a building block.

    AnExponentis a powerx in an expressionof the form bx..

    Ifb>0, b1, then ( denotes not equal to)

    f(x)=bx

    is the exponential function with base b.

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    Exponent Two category of exponential function

    b>1 0

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    Exponent

    Exponent Laws1. bm.bn= bm+n

    2. bm/bn= bm-n

    3. (bm)n= bm.n

    4. (b.c)m=bm.cm

    5. (b/c)m=bm/cm

    6. b-m=1/bm

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    The number e

    Named after Swiss mathematician Leonard Euler.

    Important in many scientific calculations.

    It is a natural base to the logarithm.

    Irrational number.Many mathematical expression

    to approximate e. Easiest expression that approximates e is the Limit

    of (1+1/x)x ,asx approaches infinity

    The value ofe to 9 decimal places:- 2.718281828

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    The number e

    200 400 600 800 1000

    2.45

    2.55

    2.6

    2.65

    2.7

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    The numbere

    Exponential function with base e. (i.e. ex )

    -10 -5 5 10

    250

    500

    750

    1000

    1250

    1500

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    Problems in Real life

    How many bits are required to represent thea discrete system of which has N levels?

    What is the minimum height of a binary treewith N nodes?

    How do I calculate complex multiplicationquickly without using a calculator to a

    reasonable accuracy.

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    Solution

    All these problems can be solved bymodeling these as logarithmic functions.

    Let us learn some formal theories relating to

    Logarithm.

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    Logarithm

    The exponenty to which a fixed number bmust be raised to produce a given numberx.

    logbx =y (1)Where b>0, b1 andx>0

    The expression (1) can be writtenequivalently as.

    by

    =x (2)

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    Logarithm

    The number b is called the base of thelogarithm.

    The numberx is the argument. It is illegal togive argument as negative or 0.

    200 400 600 800 1000

    -4

    -2

    2

    4

    6

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    Logarithm 0

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    Logarithm

    200 400 600 800 1000

    -4

    -2

    2

    4

    6

    0.2 0.4 0.6 0.8 1

    -10

    -8

    -6

    -4

    -2

    20 40 60 80 100

    -6

    -4

    -2

    2

    4

    2 4 6 8 10

    -8

    -6

    -4

    -2

    2

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    Logarithm

    Generally Logarithms with three bases areused significantly in computing procedures.

    Natural Logarithms with base as number e.

    Logarithm with base 2.

    Common Logarithm with base 10.

    We will discuss only Natural Logarithm and

    leave others as an exercise.

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    Natural Logarithms

    Logarithm with base e.

    The number system we use is of base 10. Then

    why logarithm with base e is called Natural?

    Twofold answer:- Can defined quiet easily using a simple integral or

    Taylor series.

    In real life experimentations, the expressions involving

    exponents ofe occur more often than one with base 10.

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    Natural Logarithm

    0.5 1 1.5 2 2.5 3

    0.5

    1

    1.5

    2

    e

    1/x

    =ln[e]

    And

    =ln[x]

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    Logarithm

    Natural Logarithm of function can calculatedusing a logarithmic table.

    Logarithms with other base b can also becalculated easily using the same table using astandard conversion formula

    logen=logbn/logbe

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    Logarithm with base b

    Laws of Logarithms1. logb[x.y]=logbx+logby

    2. logb[x/y]=logbx-logby

    3. logbxn=n .logbx

    4. logbx= logcx/ logcb

    logbx5. x=b

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    Nested Logarithm

    log[log[..log[n].]]

    2 4 6 8 10

    -8

    -6

    -4

    -2

    2

    ln[ln[ln[x]]]

    ln[ln[x]]

    ln[x]

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    Comparison

    Exponential functions grow very fast. Logarithmic functions grow slow & steady.

    -6 -4 -2 2 4 6-5

    5

    10

    15

    20 y=ex

    y=x

    y=ln[x]

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    Comparison

    It is desirable that various operations indifferent computer algorithms be in theorder of logarithmic complexity.

    20000 40000 60000 80000 100000

    2

    4

    6

    8

    10

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    Summary

    Computational process is expensive. Order of time and space complexity is an

    important consideration.

    Generally an algorithm with logarithmic

    complexity are considered good.