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Logarithm: It is the power to which a number must be raised in order to get some other number.
Common logarithm:Logarithms with base 10 are common logarithms.
Common logarithms are written as log10 x and if any expression is not indicated with a base, then base 10 is considered.
Natural logarithm: Logarithms with base e are natural logarithms.
Natural logarithms are written as logex and denoted as ln (x).
Logarithm of a number contains 2 parts: Characteristic and Mantissa
Characteristic is an integral part of logarithm.
Case 1: If number is greater than 1.
In this condition, characteristic is considered as one less than the number of digits in the left of decimal point in the given number.
Example: 256.23
Number of digits to the left of decimal point are 3. Hence, the characteristic is one less than number of digits before decimal points i.e 2.
Case 2: If number is less than 1.
In this condition, characteristic is considered as one more than the number of zeros between decimal point and first digit of the number. It is negative and is denoted as (One bar) 1 or (Tow bar) 2
Example: 0.00735
Number of zeros between decimal point and first significant digit 7 are 2. Hence the characteristic is one more than number of zeros i.e 3
Some important results:
(a) m =a log am
(b) x =log a (ax)
(c) log a 1 = 0
3.Some important theorems:
(a) log a (mn) = log a m + log a n
(b) log a (m/n) = log a m – log a n
(c) log a (mn) = n. log a m
(d) log a m = (log b m) / (log b a) ……. Change of base theorem
(e) log a a = 1
(f) log a b * log b a = 1


























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