Assistant professor
M.Sc. (physics)
Material science
HOW TO USE LOGS CALCULATIONS WITH ILLUSTRATIONS
Logarithms tables are useful in tedious calculations involving division, multiplication and powers of numbers.
Two types of logarithm tables
(i) Common logarithms (ii) Natural logarithm
If x and m are positive real numbers , x equal not one and y is a real number xʸ = m , then y is called the logarithm.
logₓ m = y
where, m is number ; x is base
From this definition, following results :
(i) xº = 1 .·. logₓ1= 0
(ii) x¹ = x .·. logₓ x= 1
(iii) xʸ = m ...(equation 1)
logₓ m = y ...(equation 2)
from equation (1) and (2)
we have xˡᵒᵍₓᵐ = m
(iv) logₓ (1/m) = -logₓ m
(v) logₓ c = 1/ log꜀ˣ
(vi) log 0 is not defined
Laws of Logarithms:
(i) logₓ mn = logₓ m + logₓ n
(ii) logₓ (m/n) = logₓ m - logₓ n
(iii) logₓ (mⁿ) = n logₓ m
(iv) log꜀ m = logₓ m / logₓ c ......(rule for change of base)
Common logarithms :
Logarithms of the base 10 are called common logarithms.
which is not an integral power of 10.
Logarithm of number is composed of two parts (i) characteristic(called integral part) and (ii) mantissa (called fractional part).i.e. log₁₀ m = (Characteristic) + (Mantissa).
Characteristic of log m :
(i) If m > 1, the characteristic of log m is = ( Number of digits in the integral part of m) - 1
For example :
(i) Number m = 7526 Characteristic of log m = 4-1=3
(ii) Number m = 752.6 Characteristic of log m = 3-1 = 2
(iii) Number m = 75.26 Characteristic of log m = 2-1 = 1
(iv) Number m = 7.526 Characteristic of log m = 1-1 = 0
Question : Find the characteristic of logarithm of following numbers :
0.000462
Ans : In 0.000462, there are 3 zero between decimal point and first significant digit.
So, characteristic of its logarithm will be –(3+1) = −4
(ii) If m < 1 , (i.e. 0 < m < 1), the characteristic is negative and is given by -(q+1) where q= number of zeros between the decimal point and the first non zero digit of the number when expressed in decimal form.
For example:
(i) Number m = 0.7526 Characteristic of log m = -(0+1)= -1
(ii) Number m = 0.07526 Characteristic of log m = -(1+1) = -2
(iii) Number m = 0.007526 Characteristic of log m = -(2+1)= -3
Mantissa of log m:
The mantissa part is found by using the logarithms tables, which have only four digits(is called four significant digits). The mantissa is the fractional part of a common logarithm (that is, the base 10 logarithm), the given number but not its order of magnitude.
log m = characteristic + mantissa
For example, the mantissa of both
log 30 = 1(characteristic) + 0.4771 (mantissa from logarithms table) = 1.4771
log 300 = 2.4771
when you number has more than 4 digits, consider the fifth digit. If fifth digit is 5 or more than 5,then add 1 to the fourth digit and neglect all the digits from fifth onwards.
For example :
If m > 1
(1) Number m = 120.205 ,m reduced to four significant digits 120.2, mantissa of number log m = log120.2 = 2 (Characteristic) + 0.0799 (mantissa) = 2.0799
(2) Number m =120.044 ,m reduced to four significant digits 120.0, mantissa of number log m = log120.0 = 2 (Characteristic) + 0.0792 (mantissa) = 2.0792
(3) Number m =120.045 , y reduced to four significant digits 120.1, mantissa of number log m = log120.1 = 2 (Characteristic) + 0.0795 (mantissa) = 2.0795
(4) Number m = 12.0045 , m reduced to four significant digits 12.01, mantissa of number log m = log12.01 = 1 (Characteristic) + 0.0795 (mantissa) = 1.0795
For example :
If m < 1
(1) Number m = 0.1202 , mantissa of number log m = log 0.1202 = -1 (Characteristic) + 0.0799 (mantissa) = -1.0799 (or bar1.0799)
(2) Number m = 0.01202 , mantissa of number log m = log 0.01202 = -2 (Characteristic) + 0.0799 (mantissa) = -2.0799
(3) Number m = 0.001202 , mantissa of number log m = log 0.001202 = -3 (Characteristic) + 0.0799 (mantissa) = -3.0799
NOTE: To find the log of a single digit number say 4, find the mantissa of 40.
log 4 = 0 (Characteristic) + 0.6021 (mantissa) = 0.6021
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