Usage of logarithms

Use of Logarithm Tables

Murali Burra

March 21, 2013

Murali Burra Use of Logarithm Tables

Topics to be covered



1 Introduction.



1 Introduction.
2 System of logarithms.



1 Introduction.
3 Characteristic and mantissa.



1 Introduction.
4 Two rules to ﬁnd the characteristic.



1 Introduction.
5 Tables of logarithms.



1 Introduction.
6 Determination of Mantissa from Tables..



1 Introduction.
6 Determination of Mantissa from Tables..
7 Antilogarithms.


Introduction


Introduction

Deﬁnition: If ax = N Where a> 1


Introduction

’x’ is called logarithm of N to the base ‘a’ .


Introduction


x = loga N


Introduction


x = loga N

where

ax = N −→ Exponential form or Index form.
loga N −→ Logarithmic form.


Example


Example

Index form Logarithmic form


Example

34 = 81


Example

34 = 81 log3 81 = 4


Example

34 = 81 log3 81 = 4
3−2 = 1
9


Example

34 = 81 log3 81 = 4
3−2 = 1
9 log3 ( 1 ) = −2
9


Example

34 = 81 log3 81 = 4
3−2 = 1
9 log3 ( 1 ) = −2
9

Note:
loge 1 = 0
loge e = 1


Systems of Logarithms



1 Natural logarithm:



1 Natural logarithm: Logarithms to the base ’e’ are called
natural logarithms.



natural logarithms.
Ex: loge N



natural logarithms.
Ex: loge N
2 Common logarithm:



natural logarithms.
Ex: loge N
2 Common logarithm: Logarithms to the base ’10’ are called
Common logarithms.



natural logarithms.
Ex: loge N
Common logarithms.
Ex: log10 N



natural logarithms.
Ex: loge N
Common logarithms.
Ex: log10 N
Note: When no base is mentioned it is understood to be base
’10’.


Laws of logarithms


Laws of logarithms

Product formula:

log mn = log m + log n


Laws of logarithms

Product formula:


Note:
log (m + n) = log m + log n


Laws of logarithms

Product formula:


Note:
Quotient formula:
m
log ( ) = log m − log n
n


Laws of logarithms

Product formula:


Note:
Quotient formula:
m
log ( ) = log m − log n
n
Power formula:
log mn = n log m


Characteristic and Mantissa



The integral part of the logarithm of a number is called the
’characteristic’




The positive decimal part is called ’mantissa’





Note: Characteristic may be positive, negative or zero but
’mantissa’ is always positive.





Note: Characteristic may be positive, negative or zero but
’mantissa’ is always positive.

EX:
log N = 3.6741 = 3 + 0.6741
Then characteristic is 3 and mantissa is 0.6741.



EX: If
log N = −3.6741
= −3 − 1 + 1 − 0.6741
= −4 + 0.3259
¯
= 4 + 0.3259

Then characteristic is ’-4’ and mantissa is 0.3259.


Two rules to ﬁnd the characteristic



Rule 1



Rule 1
The characteristic of the logarithm of any number > 1 is +ve.



Rule 1
(one less than the no. of digits in its integral part)



Rule 1
Ex:

324.7



Rule 1
Ex:

324.7
324.7>1



Rule 1
Ex:

324.7
324.7>1
the no. of digits in its integral part is 3.



Rule 1
Ex:

324.7
324.7>1
the no. of digits in its integral part is 3.
characteristic of 324.7 is 3-1=2


Two rules to ﬁnd characteristic

Rule 2



Rule 2
The characteristic of the logarithm of any number < 1 is -ve.



Rule 2
(Numerically equal to one more than the no. of zeros immediately
after decimal point)



Rule 2
Ex: 0.006743



Rule 2
Ex: 0.006743
0.006743<1



Rule 2
Ex: 0.006743
0.006743<1
the no. of zeros immediately after decimal point is 2.



Rule 2
Ex: 0.006743
0.006743<1
the no. of zeros immediately after decimal point is 2.
characteristic of 0.006743 is -(2+1)=-3= ¯
3


tables of logarithms

The mantissa of logarithm of a no. is calculated from the table of
logarithms.



logarithms.
these tables are four ﬁgure tables



logarithms.
If a no. consists of 5 digits??
• If the ﬁfth ﬁgure is less than 5



logarithms.
leave it.



logarithms.
leave it.
• If the ﬁfth ﬁgure is greater than 5



logarithms.
leave it.
leave it



logarithms.
leave it.
leave it
But add 1 to the fourth ﬁgure.



logarithms.
leave it.
leave it
But add 1 to the fourth ﬁgure.
If a no. consists of less than 4 digits add zeros to it till it has
four digits.


To ﬁnd mantissa from tables



1 Remove the decimal from the given no. and make it a no. of 4
digits.



digits.
2 Find ﬁrst two digits in the extreme left column of the table.



digits.
3 Trace the third digit in the top horizontal row.



digits.
Mark the no. at the intersection of the column and the row
under consideration



digits.
under consideration
4 Trace the fourth digit from the left in the column of mean
diﬀerences.



digits.
under consideration
4 Trace the fourth digit from the left in the column of mean
diﬀerences.
add the values of step(3) and (4).


example


example

Ex:
Find the logarithm of 0.005269?


example

Ex:
characteristic of 0.005269 = ¯
3


example

Ex:
3
Removing the decimal we have to ﬁnd the mantissa of 5269.


example

Ex:
3
6 9
52 7210 7


example

Ex:
3
6 9
52 7210 7
adding 7 to 7210 we gwt 7217.


example

Ex:
3
6 9
52 7210 7
preﬁxing the decimal. mantissa = 0.7217


example

Ex:
3
6 9
52 7210 7
preﬁxing the decimal. mantissa = 0.7217
log 0.005269 = ¯
3.7217


Antilogarithms


Antilogarithms

log N = x, then N= antilogx


Antilogarithms

while ﬁnding algorithms we take into account only the mantissa.


Antilogarithms

The characteristic is used only to determine the number of digits in
the integral part or the no. of zeroes immediately after the decimal.


Antilogarithms

Ex:Find antilog of 1.6975


Antilogarithms

the characteristic =


Antilogarithms

the characteristic = 1


Antilogarithms

the no. of digits in the integral part = 1+1=2


Antilogarithms

we ﬁnd the no. whose mantissa is 0.6975.


Antilogarithms

it is found to be 4983


Antilogarithms

it is found to be 4983
hence antilog 1.6975 = 49.83


Usage of logarithms

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