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How can you shorten the subject ? That stern struggle with the multiplication table, for many people not yet ended in victory, how can you make it less? Square root, as obdurate as a hardwood stump in a pasture nothing but years of efforts can extract it. you can”t hurry the process or pass from the arithmetic to algebra, you can’t shoulder your way past quadratic equations or ripple through the binomial theorem. Instead , the other way ; your feet are impeded in the tangled growth, your pace slackens, you sink and fall somewhere near the binomial theorem with the calculus in sight on horizon. So died, for each of us, still bravely, lighting our mathematical training; except for a set of people called ” mathematician” – born so , like crooks.
                                                                                                                         – Stephen Leacock

The word Logarithm is the combination of two words- Logos and arithmos. The meaning of these two Greek words are ration and number respectively. Hence logarithm means ration number.  During the later part of 16th century, sea voyages were taken in large scales by the western countries like Great Britain, Portugal and Denmark. So the correct position of stars, planets and constellation were needed and therefore it was necessary to prepare accurate trigonometric tables for which complicated calculations were necessary.
In 1593, the great work of two Danish mathematicians Wittich and Calvin’s De – Astrolabo was published, who suggested the use of trigonometric table for shortening calculation. Moreover, Stevinus had published a table for calculations in commercial mathematics, which helped the person taking voyages to calculate the wealth collected during the voyages.
The birth of logarithm was therefore to shorten the length of calculation.
Take one example –
256 + 225 = 481
256 x 225 = 57600 and it involves three steps.
It is clear from the examples that number of operations involved in multiplication is greater than the number of operations involved in adding them. The bigger the number is the amount of labor in involved in calculation. Hence an effort was made to reduce all multiplication or division into addition or subtraction problems, and thus had the birth of logarithms possible.
John Napier is called the father of logarithms. John Napier was born in Scotland in 1550. Although he was not a professional mathematician but he had a strong interest in simplifying calculations, he worked for 20 years to prepare the table of logarithms. Napier published Mirfici Logarithmorum canonies Descripto in 1614 which was translated in English by Edward Wright.
John Napier
Napier approached logarithm form the stand point of geometry and probably he might have used the formula –

2 Sin A Sin B = Cos(A – B) – Cos(A + B)
To prepare the logarigthm table. But presently we take
am x an = am + n
to understand the log. The fact that in Napier’s log table the value of log1 does not equal to zero brought a major difficulty in its use. In a meeting of 1615, Napier suggested Briggs to construct a log table with a base 10 and with log1 = 0.  Henry Briggs, a professor of geometry at Gresham College, London upon reading the Napier’s Descripto wrote –

Napier, Lord of Mar Kinston, hath set my head and hands at work with his new and admirable logarithms. I hope to see him this summer, if it pleases God; for I never saw a book which pleased me better and made me more wonder.
Prof. Briggs in his Arithmetica Logarithmica writes about Napier in the following words –
I journeyed to Edinburgh where being most hospitably received by him, I lingered for a whole month. But as we talked over the changes in logarithm, he said that he had for some times been of the same opinion and had wished to accomplish it …. He was of the opinion — that 0 should be the logarithm of unity.
Napier in his book Mirfici Logarithmorum canonies Descripto wrote  a work on the construction of a table published in 1619 posthumously. Prof. Briggs in 1624 published his Arithmetica Logarithmica gave the logarithms of all numbers from 1 to 1000 correct to 14 decimal places.
We all know
2 Sin A Cos B = Sin(A+B) + Sin(A- B)
Suppose we have to multiply — 0.7072 x 0.9781
From the trigonometric table
Sin45 = 0.7072                                    and Cos12 = 0.9781
Hence,
Sin45 x Cos12 = ½ (Sin57 + Sin33)
Sin 57 = 0.8387                                               and  Sin33 = 0.5446
Hence,
Sin45 x Cos12 = ½ (0.8387 + 0.5446) = 0.69165
By general multiplication – 0.7072 x 0.9781 = 0.69171232
If we compare both the result we see both the results are true to the 3 decimal places. The inaccuracy is due to the fact the table used here is for four-figure table. Had we consulted the eight figure table we would have the seven figure correct answer. 
The present day log table is based on indices-
If
ax  = n
log an = x
n = antilog ax
here the operator log written in front of number means – Look up in the table the power to which a  has to be raised to give the number , whereas operator Antilog says – Look up in the table the value of the base when raised to the power represented by the number.
How can you construct a log table by your own?
We know
100 = 1
log101 = 0
210 = 1024 = 103 approximately
Since 1024 – 1000 = 24 , i.e a difference of 2 ½ %
Hence,
2 = (103 )1/10 = 100.3 approximately
Therefore, log 102 = 0.3
Similarly,
Since 39 = 19683 = nearly equal to 20000
= 2 x 10000 approximately
= 100.3 x 104 = 104.3   approximately
3 = (104.3)1 / 9 = 100.48
log103 = 0.48 
In the same way you can prepare a log table.
Rule of log:-
Check some more rules on logarithms

Rajesh Kumar Thakur
rkthakur1974@gmail.com

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