Today our topic of discussion is Characteristics of Common Log.
Characteristics of Common Log
Characteristics of Common Log
Let a number N be expressed in the scientific form as
N = a * 10 n where N > 0, 1 <= a < 10 and n in Z
Taking base 10 log of both sides we get.
log_10(N) = log_10(a * 10n)
= log_10(a) + log_10(10 n)
= log_10(a) + nlo*g_{10} * 10 .. log 10 N
=n+log 10 a[** log_10(10) = 1 ]
Suppressing the base 10, we have,
log(N) = n + log(a)
n is called the characteristic of log(N)
Note: It is evident from the table that characteristic of a number having m digits to the left of decimal point is 1 less than the number of digits to the left of decimal sign and that will be positive. That is if the mentioned number of digits is m, the characteristic of logarithm will be m-1.
Note: Now observe from the following table: If there is no integral part of a number and the number has k 0’s immediately after decimal point, then the characteristic of the logarithm of the number is \{- (k + 1)\} .
If any characteristic is negative, sign is not placed on the left of the characteristic, rather (bar sign) over the characteristic. Such as, characteristic -3 will be written as 3. Otherwise, whole part of the log including mantissa will be negative.
Note: Characteristic may be either positive or negative, but mantissa will always be positive.
Example 11. Find the characteristics of log of the following numbers:
1) 5570
2) 45.70
3) 0.4305
4) 0.000435
Solution:
1) 5570 = 5.57 * 1000 = 5.57 * 103
∴ Characteristic of log of the number is 3
Otherwise, the number of digits in the number 5570 is 4.
∴ Characteristic of log of the number is = 4-1-3
2) 45.7 = 4.57 * 10 ¹
∴Characteristic of log of the number is 1
Otherwise, there are 2 digits in the integral part (i.c. on the left of the decimal point) of the number.
∴Characteristic of log of the number is = 2-1 = 1
3) 0.4305 = 4.305 * 10 ^ – 1 .
Characteristic of log of the number is -1
∴ Characteristic of log of the number is = 0 – 1 = – 1 = overline 1
Otherwise, there is no zero in between decimal point and its next first significant digit of the number 0.4305, i.e. there is 0 zeroes.
∴ Characteristic of log of the number is = – (0 + 1) = – 1 = overline 1
∴ Characteristic of log of the number 0.4305 is Ï
4) 0.000435 4.35 × 10-4
∴ Characteristic of log of the number is -4 or 4 Otherwise, there are 3 zeroes in between decimal point and its next first significant digit of the number 4.
∴ Characteristic of log of the number is (3+1)=-4-4 =
∴Characteristic of log of the number 0.000435 is 4
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