# exponent logarithm

Post on 07-Apr-2018

226 views

Category:

## Documents

Embed Size (px)

TRANSCRIPT

• 8/6/2019 Exponent Logarithm

1/25

CDAC Mumbai 1

Exponent and Logarithm

• 8/6/2019 Exponent Logarithm

2/25

CDAC Mumbai 2

Overview

Exponent Logarithm

• 8/6/2019 Exponent Logarithm

3/25

CDAC Mumbai 3

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.

• 8/6/2019 Exponent Logarithm

4/25

CDAC Mumbai 4

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.

• 8/6/2019 Exponent Logarithm

5/25

CDAC Mumbai 5

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.

• 8/6/2019 Exponent Logarithm

6/25

CDAC Mumbai 6

Exponent Two category of exponential function

b>1 0

• 8/6/2019 Exponent Logarithm

7/25

CDAC Mumbai 7

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

• 8/6/2019 Exponent Logarithm

8/25

CDAC Mumbai 8

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

• 8/6/2019 Exponent Logarithm

9/25

CDAC Mumbai 9

The number e

200 400 600 800 1000

2.45

2.55

2.6

2.65

2.7

• 8/6/2019 Exponent Logarithm

10/25

CDAC Mumbai 10

The numbere

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

-10 -5 5 10

250

500

750

1000

1250

1500

• 8/6/2019 Exponent Logarithm

11/25

CDAC Mumbai 11

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.

• 8/6/2019 Exponent Logarithm

12/25

CDAC Mumbai 12

Solution

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

Let us learn some formal theories relating to

Logarithm.

• 8/6/2019 Exponent Logarithm

13/25

CDAC Mumbai 13

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)

• 8/6/2019 Exponent Logarithm

14/25

CDAC Mumbai 14

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

• 8/6/2019 Exponent Logarithm

15/25

CDAC Mumbai 15

Logarithm 0

• 8/6/2019 Exponent Logarithm

16/25

CDAC Mumbai 16

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

• 8/6/2019 Exponent Logarithm

17/25

CDAC Mumbai 17

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.

• 8/6/2019 Exponent Logarithm

18/25

CDAC Mumbai 18

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.

• 8/6/2019 Exponent Logarithm

19/25

CDAC Mumbai 19

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]

• 8/6/2019 Exponent Logarithm

20/25

CDAC Mumbai 20

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

• 8/6/2019 Exponent Logarithm

21/25

CDAC Mumbai 21

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

• 8/6/2019 Exponent Logarithm

22/25

CDAC Mumbai 22

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]

• 8/6/2019 Exponent Logarithm

23/25

CDAC Mumbai 23

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]

• 8/6/2019 Exponent Logarithm

24/25

CDAC Mumbai 24

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

• 8/6/2019 Exponent Logarithm

25/25

CDAC Mumbai 25

Summary

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

important consideration.

Generally an algorithm with logarithmic

complexity are considered good.