Why the concept was discovered/invented?
To simplify calculations of very large numbers
Anything related to logarithm was first published in 1614, by John Napier. At that time, the most troublesome thing to mathematical practice was Arithmetic of very large numbers. John Napier was a hobby astronomer as well and he was aware that Astronomical calculations required the multiplication and division of very large numbers, something that’s pretty hard to do without a calculator. Looks like that Napier invented logarithms to aid calculation in astronomy.
How did it help human society at that time ?
The problem of multiplying lots of numbers was the original reason logarithms were developed. The logarithmic system was developed independently by the Scottish mathematician John Napier & a Swiss clockmaker Joost Burgi, almost simultaneously & parallely the in early 17th century and later refined to their present form with aid of Henry Briggs, another English mathematician. This was primarily in response to large cumbersome computations which were necessitated in several key scientific areas of that time, such as in observational astronomy, navigation and geodesy and others.
How is it helping human society now ?
Thought process/Intuition behind the discovery/invention.
If you have to do
10000 x 100000
Don’t you just count the number of zeros and add them
10000 x 100000 = 1000000000
10 power 4 x 10 power 5 = 10 power 9
Lets see if this logic works for numbers other than 10 as well
27 x 81 =
3pow3 x 3pow4 = 3pow7
In Napier’s time, however, people were not used to thinking in terms of exponentiation. They didn’t have the concept of a base and they didn’t have our handy way of writing powers, using a little number at the top.
After we raise & answer these questions, we should start telling about any terminology/technique/formulae for any concept.
Tables of numbers related in a very similar way were first published in 1614 by the mathematician, physicist and astronomer John Napier in a paper called The construction of the wonderful canon of logarithms.
Logarithms were invented independently by John Napier, a Scotsman, and by Joost Burgi, a Swiss. The logarithms which they invented differed from each other and from the common and natural logarithms now in use. Napier’s logarithms were published in 1614; Burgi’s logarithms were published in 1620. The objective of both men was to simplify mathematical calculations. Napier’s approach was algebraic and Burgi’s approach was geometric. Neither men had a concept of a logarithmic base. Napier defined logarithms as a ratio of two distances in a geometric form, as opposed to the current definition of logarithms as exponents. The possibility of defining logarithms as exponents was recognized by John Wallis in 1685 and by Johann Bernoulli in 1694.
Logarithms are useful in many fields from finance to astronomy.
The idea was to simplify multiplicaton of numbers. If you ever tried to multiply 1010-digit numbers by hand you will see what I am talking about.
What was Napier’s purpose for inventing this system? He developed this shortcut to save astronomers time and limit “slippery errors” of calculations. His idea was that by “shortening the labours, doubled the life of the astronomer.” With his logarithms, Napier presented a mechanical means of simplifying calculations in his Rabdologiae in 1617. He described a method of multiplication using rods with numbers marked off on them. This was the earliest form of mechanical calculation and the forerunner of our modern day calculator.
Napier, who is credited with the invention of logarithms, only considered the study of mathematics as a hobby.
Napier not only “invented” logarithms, he had many other achievements in his lifetime. Among these accomplishments were revolutionary methods for tilling and fertilizing soil and a number of “Secret Inventions” to defend his country from Philip of Spain. These inventions include the round chariot with firepower and heavy protection (the idea behind the tank); an underwater ship (the submarine); and an artillery piece which would mow down a field of soldiers (the machine gun). It is rumored that while testing his machine gun, he took out an entire flock of sheep in a pasture outside of his estate. After witnessing the destructive power his invention possessed, he swore never to make another gun or give the information on how to make one to anyone.
John Napier invented logarithms, but many other scientists and mathematicians helped develop Napier’s logarithms to the system we use today. The first table of common logarithms was compiled by the English mathematician Henry Briggs. In 1624, while working with Napier, Briggs and Napier discovered natural logarithms which first arose as more or less “accidental variations” of Napier’s original logarithms. The possibility of defining logarithms as exponents was recognized by John Wallis in 1685 and later by Johann Bernoulli in 1694.
Timeline of Logarithms
by Anthony Fogleman
| 1550 | John Napier1 was born in Edinburgh Scotland. |
| 1552 | Jobst Bürgi was born in Switzerland. |
| 1588 | Bürgi began working on his logarithms2 independent of Napier (I was unable to find the base to which Bürgi created his logarithms). |
| ~1594 | John Napier started work on his tables and spent the next twenty years completing. The tables were for trigonometric applications and gave the logarithms for the sine of angles 30o to 90o. Although Napier did not actually use in his logarithms it could be said his base was roughly 1/e. |
| 1614 | Napier published “Mirifici logarithmorum canonis descriptio” in which he discusses his logarithms. |
| 10 March 1615 | Henry Briggs wrote a letter roughly translating questions Napier’s use of his base (1/e) and why he did not use base 10 and log 1 = 0. Napier replied that he too had the idea but could not create the tables due to an illness. |
| Summer 1615 | Henry Briggs visited John Napier and they spent a month working on the tables for the logarithms to base 10. |
| 1616 | Henry Briggs visited John Napier a second time. |
| 4 April 1617 | John Napier passed away. |
| 1617 | Briggs published his “Logarithmorum Chilias Prima” which contained his tables for logarithms to base 10. |
| 1619 | “Mirifici logarithmorum canonis constructio” is published in which the method Napier used for constructing his logarithms is discussed. |
| 1620 | Bürgis’ were published in his “Arithmetische und Geometrische Progress- Tabulen.” Bürgi’s work went unnoticed due to the beginning of the Thirty Years’ War. |
| 1622 | William Oughtred invented the slide rule, which offered an even quicker way of calculating logarithms. |
| 1632 | Jobst Bürgi passed away. |
| 1675 | Newton discovers the fact that the d/dx ln x = 1/x. |
| 1685 | John Wallis realized that logarithms could be defined as exponents. |
| 1694 | Johann Bernoulli also realized that logarithms could be defined as exponents. |
| 1694 to present | Logarithms had reached their full potential and most of what was done after 1694 was calculating logarithms to different bases. |
https://www.thocp.net/reference/sciences/mathematics/logarithm_hist.htm
https://jscholarship.library.jhu.edu/bitstream/handle/1774.2/34187/31151005337641.pdf
A modern Mathematician regards the logarithmic function as the inverse of an exponential function; and it may seem to us, familiar as we all are with the use of operations involving indices, that the conception of a logarithm would present itself in that connection as a fairly obvious one. W e must however remember that, at the time of Napier, the notion of an index, in its generality, was no part of the stock of ideas of a Mathematician, and that the exponential notation was not yet in use.
It was in the early years of the seventeenth century that Johannes Kepler was engaged in the prodigious task of discovering, and verifying by numerical calculation, the laws of the motion of the planets. In this age of numerical calculation then Napier occupied himself with the invention of methods for the diminution of the labour therein involved.
In particular the publication of the wonderful canon was received by Kepler with marked enthusiasm. In his “Ephemeris” for 1620, Kepler published as the dedication a letter addressed to Napier, dated July 28, 1619, congratulating him warmly on his invention and on the benefit he had conferred upon Astronomy.
In a letter to Archbishop Ussher, dated Gresham House, March 10, 1615, Briggs wrote, ” Napper, lord of Markinston, hath set m y head & hands a work with his new & admirable logarithms. I hope to see him this summer, if it please God, for I never saw book which pleased me better, or made m e more wonder.” Briggs visited Napier, and stayed with him a month in 1615, again visited him in 1616, and intended to visit him again in 1617, had Napier’s life been spared.
It must be remembered that, at that time and long afterwards, the sine of an angle was not regarded, as at present, as a ratio, bu{ as the length of that semi-chord of a circle of given radius which subtends the angle at the centre. (History of Trignometry)
Napier had no explicit knowledge of the existence of the number e, nor of the notion of the base of a system of logarithms, although as we shall see he was fully cognizant of the arbitrary element in the possible systems of logarithms.
Our present notation for numbers with the decimal point appears to have been independently invented by Napier, although a point or a half bracket is said to have been employed somewhat
earlier by Jobst Biirgi for the purpose of separating the decimal places from the integral part of a number*.
Decimal Point – The invention of decimal fractions was due to Simon Stevin (i 548-1620), who published a tract in Dutch, ” D e Thiende,” in 1585, and in the same year one in French under the title ” La Disme,” in which the system of decimal fractions was introduced, and in which a decimal system of weights, measures and coinage was recommended. In the “Rabdologia” Napier refers to Stevin in an ” Admonitio pro Decimali Arithmetica ” in which he emphasizes the simplification arising from the use of decimals, and introduces the notation with the decimal point. By Stevin and others the notation 94 ® 1 © 3 © o (3) 5 © or 94 1′ fom5″”, for example, was used instead of Napier’s notation 94″ 1305. Later on Briggs sometimes used the notation 941305. It is clear that the notation introduced by Napier, which was however not * Th e decimal point was also employed by Pitiscus in the tables appended to the later edition, published in 1612, of his ” Trigonometria.” universally adopted until the eighteenth century, w a s far better adapted than the mor e complicated notation used b y Stevin an d later writers to bring out the complete parity of the integral an d decimal parts of a numbe r in relation to the operations of arithmetic, and to emphasize the fact that the system of decimal fractions involves only an extension of the fundamental conception of our system of notation for integral numbers , that the value of a digit, in relation to the decimal scale, is completely indicated b y its position.
https://plus.maths.org/content/dynamic-logarithms
http://mathforum.org/mathimages/index.php/The_Logarithms,_Its_Discovery_and_Development
https://www.maa.org/press/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-john-napier-introduces-logarithms
Log
Matrix
Statistics
What is a solution?
Website Themes
Baskerville2
Rebalance
Argent
When the numbers are too big or too small. For instance, pH is a more “manageable” number than the corresponding concentration of hydronium ions in an acid.
https://betterexplained.com/articles/using-logs-in-the-real-world/
http://www.sosmath.com/algebra/logs/log1/log1.html
https://plus.maths.org/content/dynamic-logarithms
