Mathematics in Ancient and Medieval India c5.s4: Logarithms
Mathematics in Ancient and Medieval India c5.s4
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Logarithms
c5.s4.1
The expression log ₂ 8 = 3 means the number 8 can be divided by 2 three times. This was defined by Napier (1550–1617) who has been given priority with regard to the publication of a table of logarithms and an account of their meaning and use. In the earliest centuries of the Christian era Jainas (vide *Dhavalā* commentary of Vīrasenācārya) conceived the idea of logarithms to the base 2, 3, 4, though no general use of the idea seems to have been made by them. They used *ardhaccheda*, *tṛkaccheda*, *caturthaccheda* of a number, as the number of times it can be divided by 2, 3, 4, etc. For instance *ardhaccheda* of x = log ₂ x, *tṛkaccheda* of x = log ₃ x etc. Moreover, there is evidence in the *Dhavalā*² commentary that the following rules were known to them.
c5.s4.2
² Singh, A. N. ‘History of Mathematics in India from Jaina Sources’, *Jaina Antiquary*, 15, no. 2, 46-53, 1949.
c5.s4.3
(1) log m/n=log m-log n
c5.s4.4
(2) log (m. n)=log m+log n
c5.s4.5
(3) log ₂ (2)m=m
c5.s4.6
(4) log (xx)=2x log x
c5.s4.7
(5) log log (xx)=log x+1+log x; the expression log log x is known as vargasalaka of x.
c5.s4.8
(6) log (xxx)=xxlog xx etc.