# exponent logarithm

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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.