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Session

Logarithms

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Session Objectives

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Session Objectives

1. Definition

2. Laws of logarithms

3. System of logarithms

4. Characteristic and mantissa

5. How to find log using log tables

6. How to find antilog

7. Applications

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Base:Any postive real number

other than one

Logarithms Definition

alog N x

Log of Nto the

base a is x

xa Nalog N x

2

2Example : log 4 2 2 4 Note: log of negatives andzero are not Defined in Reals

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Illustrative Example

The number log27 is

(a) Integer (b) Rational

(c) Irrational (d) Prime

Solution:

Log27 is an Irrational number

Why?

As there is norational number,

2 to the powerof which gives 7

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Fundamental laws of logarithms

b b b1) log xy log x log y

b bLet log x A, log y B

A Bb x , b y

A B A Bxy b b b

b b blog xy A B log x log y hence proved

b b bx2) log log x log yy

y

b b3) log x y log x

b b b bExtension log xyz log x log y log z

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Other laws of logarithms

0

b4) log 1 0 as b 1

1

b5) log b 1 as b b

ab

a

log x6) log x

log b

Changeof base

blog x7) b x

blog xLet b y blog x

b blog b log y

b b blog xlog b log y b blog x log y

y x

z

y

bb

y8) log x log x

z

Where a is any other base

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Illustrative Example

2

3

Simplify log 2 2

Solution:

2

3 2log 2 2 log 2 23

3

22

log 23

2 3

. log 2 log 23 2

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Illustrative Example

Solution :

log 7 log 33 7

3 7 True / False ?

log 73

log 7 log 73 33 3

1

log 73 log 733

1

log 7 log 33 7

7 7

Hence True

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Illustrative Example

Solution:

If ax= b, by= c, cz= a, then the

value of xyz is

a) 0 b) 1 c) 2 d) 3

xa b xloga logb

logbx

loga

logc logaSimilarly y , zlogb logc

Hence xyz 1

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Illustrative Example

Find log tan 0.25

Solution:

log tan 0.25 log tan4

log 1 0

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Illustrative Example

Solution:

1 1 1log ...2.5 2 33 3 3Pr ove that 0.16 4

1 1 1 1 / 3

log ... log2.5 2.52 33 1 1 / 33 30.16 0.16

1 / 3

log2.52 / 30.16 1

2log2.5 20.4 21

log2.5 20.4

21

log10 24

4

10

21

log10 24

10

4

2

1log 210 2

410 1

44 2

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Illustrative Example

Solution:

If log32, log3(2x-5) and log3(2

x-7/2)

are in arithmetic progression, thenfind the value of x

2log3(2x-5) = log32 + log3(2

x-7/2)

log3(2x-5)2= log32.(2

x-7/2)

(2x-5)2= 2.(2x-7/2)

22x -12.2x + 32 = 0, put 2x= y, we get

y2- 12y + 32 = 0 (y-4)(y-8) = 0 y = 4 or 8

2x=4 or 8 x = 2 or 3

Why

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Illustrative Example

Solution:

If a2+4b2= 12ab, then prove that

log(a+2b) is equal to

1

loga logb 4log22

a2+4b2= 12ab (a+2b)2 = 16ab

2log(a+2b) = log 16 + log a + log b

2log(a+2b) = 4log 2 + log a + log b

log(a+2b) = (4log 2 + log a + log b)

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System of logarithms

Common logarithm:Base = 10

Log10x, also known as Briggssystem

Note: if base is not given base is

taken as 10

Natural logarithm:Base = e

Logex, also denoted as lnx

Where e is an irrational number given by

1 1 1e 1 .... ....

1! 2! n!

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Illustrative Example

Solution:

lnln7e 7 True / False ?

Hence False

log blnln7 ae ln7 as a b

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Characteristic andMantissa

Standard form of decimal

pn m 10 where 1 m 10

3Example 1234.56 1.23456 10

3

0.001234 1.234 10

p pHence log n log m 10 log m log 10

log n log m plog 10 log m p

p is characteristicof n

log(m) is mantissaof n

log(n)=mantissa+characteristic

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How to find log(n) using logtables

1) Step1: Standard form of decimal

n = m x 10p, 1 m < 10

log n p log m

Note to find log(n) we have tofind the mantissa of n i.e. log(m)

2) Step2: Significant digits

Identify 4 digits from left, starting from first nonzero digit of m, inserting zeros at the end ifrequired, let it be abcd

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How to find log(n) using logtables

n Std. form

m x 10pp m abcd

1234.56 1.23456x103 3 1.2345 1234

0.000123 1.23x10-4 -4 1.23 1230

100 1x102 2 1 1000

0.10023 1.0023x10-1 -1 1.0023 1002

Example n = m x 10p

,

p: characteristic, log(m): mantissa

Log(n) = p + log(m)

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How to find log(n) using log tables

3) Step3: Select row ab

Select row ab from thelogarithmic table

4) Step4: Select column c

Locate number at column cfrom the row ab, let it be x

5) Step5: Select column of mean difference d

If d 0,Locate number at column dof mean difference from the rowab, let it be y

What if d = 0?Consider y = 0

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How to find log(n) using log tables

6) Step6: Finding mantissa hence

log(n)

Log(m) = .(x+y)

Log(n) = p + Log(m)

Summarize:

1) Std. Form n = m x 10p

2) Significant digits of m: abcd

3) Find number at (ab,c), say x, where ab: row, c: col

4) Find number at (ab,d), say y, where d: mean diff

5) log(n) = p + .(x+y)

Never neglect 0s

at end or front

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Illustrative Example

Find log(1234.56)

n Std. form

m x 10pp m abcd

1234.56

1.23456x103

3 1.2345 1234

1) Std. Form n = 1.23456 x 103

2) Significant digits of m: 1234

3) Number at (12,3) = 0899

4) Number at (12,4) = 14

5) log(n) = 3 + .(0899+14) = 3 + 0.0913 = 3.0913

Note this

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Illustrative Example

Find log(0.000123)

n Std. form

m x 10pp m abcd

0.0001

23

1.23x10-4 -4 1.23 1230

1) Std. Form n = 1.23 x 10-4

2) Significant digits of m: 1230

3) Number at (12,3) = 0899

4) As d = 0, y = 0 Note this

5) log(n) = -4 + .(0899+0) = -4 + 0.0899 = -3.9101

To avoidthe

calculations

4.0899

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Illustrative Example

Find log(100)

n Std. form

m x 10pp m abcd

100 1x102 2 1 1000

1) Std. Form n = 1 x 102

2) Significant digits of m: 1000

3) Number at (10,0) = 0000

4) As d = 0, y = 0

5) log(n) = 2 + .(0000+0) = 2 + 0.0000 = 2

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Illustrative Example

Find log(0.10023)

n Std. form

m x 10pp m abcd

0.10023

1.0023x10-1

-1 1.0023 1002

1) Std. Form n = 1.0023 x 10-1

2) Significant digits of m: 1002

3) Number at (10,0) = 0000

4) Number at (10,2) = 9

5) log(n) = -1 + .(0000+9) = -1 + 0.0009 = -0.9991

To avoidthe

calculations

1.0009

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How to find Antilog(n)

(1) Step1: Standard form of number

If n 0, say n = m.abcd

For bar notation subtract 1, add 1 we get

If n < 0, convert it into barnotation say n m.abcd

For eg. If n = -1.2718 = -1 0.2718

n = -1-0.2718=-2+1-0.2718

n = -2+0.7282

2.7282

Now n = m.abcd or n m.abcd

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How to find Antilog(n)

2) Step2: Select row ab

Select the row ab fromthe antilog table

Eg. n = -1.2718 2.7282

Select row 72 from table

3) Step3: Select column c of ab

Select the column c ofrow ab from the antilogtable, locate the number

there, let it be x

Eg. n 2.7282

Number at col 8 of row72 is 5346, x = 5346

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How to find Antilog(n)

4) Step4: Select col. d of mean diff.Select the col d of meandifference of the row abfrom the antilog table, letthe number there be y, Ifd = 0, take y as 0

Eg. n 2.7282

Number at col 2 of meandiff. of row 72 is 2, y = 2

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How to find Antilog(n)

5) Step5: Antilog(n)

If n = m.abcd i.e. n 0

Antilog(n) = .(x+y) x 10m+1

If i.e. n < 0

Antilog(n) = .(x+y) x 10-(m-1)

n m.abcd

Eg. n 2.7