11. Logarithms

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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