the characteristic of log 3.126 is
Answers
Answer:
The characteristic of the logarithm of a number greater than 1 is positive and is one less than the number of digits in the integral part of the number. (ii) To find the characteristic of the logarithm of a number lying between 0 and 1: Since, log . 1 = -1 and log 1 = 0, hence the common logarithm of a number between.
Step-by-step explanation:
two types of logarithms are used:
(i) Natural or Napierian logarithm
(ii) Common logarithm
The logarithm of a number to the base e is known as Napierian or Natural logarithm after the name of John Napier; here the number e is an incommensurable number and is equal to the infinite series:
1 + ¹/₁₀ + ¹/₂₀ + ¹/₃₀ + ………… ∞
The logarithm of a number to the base 10 is known as common logarithm.
This system was first introduced by Henry Briggs. This type is used for numerical calculations. The base 10 in common logarithm is usually omitted.
For example, log₁₀ 2 is written as log 2.
The rest of the part deals with the method of determining common logarithms of positive numbers.
Characteristic and Mantissa:
Now, consider a number (say 6.72) between 1 and 10. Clearly,
1 < 6.72 < 10
Therefore, log 1 < log 6.72 < log 10
or, 0 < log 6.72 < 1 [ Since log 1 = 0 and log 10 = 1]
Therefore, the logarithm of a number between 1 and 10 lies between 0 and 1. That is,
log 6.72 = 0 + a positive decimal part = 0∙ …………..
We now consider a number (say 58.34) between 10 and 100. Clearly,
10 < 58.34 < 100
Therefore, log 10 < log 58.34 < log 100
or, 1 < log 58.34 < 2 [Since log 10 = 1 and log 100 = 2 ]
Therefore, the logarithm of a number between 10 and 100 lies between 1 and 2. That is,
log 58.34 = 1 + a positive decimal part = 1∙ ..
Similarly, the logarithm of a number (say 463) between 100 and 1000 lies between 2 and 3 (since log 100 = 2 and log 1000 = 3). That is,
log 463 = 2 + a positive decimal part = 2∙
In like manner the logarithm of a number between 1000 and 10000 lies between 3 and 4 and so on.