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(206) Nandanavana
However, the symbols indicate that symbolizations were in vogue even in early post-Christian era. Logarithm in Dhavala
The logarithmic process is also a method of expressing large numbers in simple forms and performing the fundamental operations through addition and subtraction. Dhavalā has utilized this process in many cases but its basis is not 10 or Naperian (log10 In or 2.303 log10). Here, the base is 2,3 or 4. Many new terms have been used in Dhavalā in this context:
1. Ardhaccheda (Ac) means log2 (log to the base 2). The Ac of a number is equal to the number of times it can be halved. 2. Varga-salākā (Vs) of a number is Ac of Ac of the number (log.log2).
3. Trika-cheda (Tc) is logarithm to the base 3. 4. Caturthacheda = log to the base 4. Thus, we have the following logarithmic formulae:
(1) Log4 2m = m;
m;
(3) Log3 3m = m;
(1) log(a/b)
(2) log ab (3) log (a)2
As per Vīrasena calculations
(4) Log4 4m = m;
This kind of logarithm is not found in contemporary Indian mathematics. This is a specialty of Jain Mathematics alone. However, this method could not be popular as no log tables were prepared by Jaina scholars like those of today. Had it been so, the calculations would have been much simpler in the past. The current logarithm was developed by Napier and Burgi (early 17th century) on the basis of 10 or 2.303 log 10 or In. From the varied descriptions in Dhavalā. We find the following logarithmic formulae:
Se
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=
=
29
log, log, x3
log, x3 = x2
<
log a - log b
log a + log b 2a log b
{x | 1}2
(2) Log2 log2 2m
-2
log, x.
=
Where x sign indicates squared-squaring. This shows that even this process could be logarithmised.
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