There has been some debate over who the original inventor of logarithms was. The two people most commonly attributed with the honor are John Napier, a Scottish Baron of Merchiston (Clark & Montelle, 2011), and Joost Burgi, a Swiss craftsman. Two others that are worth mentioning are Edward Wright, an English mathematician, and Christian Longomontanus, a Danish Astronomer. (Knott, 1915)
Benjamin Martin, an eighteenth-century writer of London, made claims on behalf of Edward Wright for the invention of logarithms. Martin explains that Wright produced a text in 1599 entitled Certaine Errors in Navigation corrected which had a Table of Latitudes included. This table contains the “numbers expressing the length of an arc of the nautical meridian” (Clark & Montelle, 2011). Or in other words, the table's values helped navigators to maintain a proper position on some chart used for navigation which had parallel longitude lines, thus creating the parallels of latitude became larger than they should be. To correct this error you must multiply “the length of the small arc by the secant of the latitude” , which was called the “nautical meridian” as mentioned above (Clark & Montelle, 2011).
The argument against Wright is that this table wasn't invented for the purpose of logarithms themselves. Martin's motivation was only to help in correcting errors in navigation. So his table was not of logarithms directly, but at most his table could be interpreted as a “table of logarithms of the tangents of half the complements of latitude” (Clark & Montelle, 2011). So most do not attribute Edward Wright as the inventor of logarithms.
The story supporting Christian Longomontanus as the inventor of logarithms is actually quite a funny one. The tale, told by some Anthony Wood, is that a college of John Napier was telling Napier about some “'new invention in Denmark (by Longomontanus as 'tis said) to save the tedious multiplication and division' involving 'proportionable numbers'” (Clark & Montelle, 2011). This idea of saving tedius multiplication and division using proportional numbers may be referring to the idea of logarithms, and apparently this conversation took place before Napier had produced any text concerning logarithms. The argument is that Longomontanus may have already been working on logarithms.
However, no works published by Longomontanus contained any trace of concepts similar to logarithms. One possible explanation is that Longomontanus was actually working on trigonometric formulas which was being studied about this same time to do the same thing.
In 1591 Burgi completed his astronomical clock. In that same year, it is suggested that he was using his own version of logarithms which he invented to help in with his astronomical calculations. “It is not clear precisely when he started using logarithms but most historians believe that he invented them around 1588” (O'Connor & Robertson, 2010).
In December 1604, Bugi moved to Prague and started working with Kepler. It was Kepler who convinced Burgi to produce a text with his work on logarithms. This work was printed in 1620, six years after Napier's publication. Both Kepler and Benjamin Bramer, Burgi's brother-in-law who lived with him from 1603 till 1611, witnessed that Burgi had completed his work on logarithms long before he compiled a text on it (Clark & Montelle, 2011). However, most people give credit to Napier because his text was published first.
Burgi invented logarithms so that he could use one single table for all arithmetical operations rather than referring to many tables for various operations. In the work published in 1920, Burgi describes having to use a variety of tables for different operations as “not alone irksome, but also laborious and cumbersome” (located in the preface of his text Arithmetische und Geometrische Progess Tabulen, sited from Clark & Montelle, 2011).
For his table, Burgi uses a common ratio of 1.0001 for precision sake. He computed logarithm values for 108 to 109, which produced a table 58 pages long with a total of 23,030 entries. In addition to this he also wrote an accompanying work with examples on how the table could be used for “all manner of calculations, including multiplication, division, extraction of roots, and computing mean proportionals.” (Clark & Montelle, 2011)
Napier also founded logarithms for the purpose of reducing computation. He understood that most “practitioners who had laborious computations generally did them in the context of trigonometry”(Clark & Montelle, 2011). Therefore, the logarithms that Napier developed were in a trigonometric context.
The theory behind the logarithms Napier discovered was based on the relation between arithmetical and geometrical progressions as well as some physics (Knott, 1915). Basically Naper imagined two particles starting in the same horizontal location and traveling along parallel lines with the same velocity. One of these lines was infinite, while the other was finite. The distance to be covered on the finite line was the sine where the distance that was covered on the infinite line was the logarithm of the sine. Thus, as the sines decreased, the logarithms increased. Below is a diagram of this relationship.
Napier published two texts concerning logarithms. In the first text Constructio published in 1614, he used the term “artificial numbers” to describe his logarithms (Shell-Gellasch, 2010). In his second text A Description of the Wonderful Table of Logarithms he coined the term logarithms. The term came from two Greek works: logos, meaning proportion, and arithmos, meaning number. So together logarithm means “proportion number” (Clark & Montelle, 2011). Napier compiled tables that contained almost ten million entries which he estimated took him roughly twenty years. This would put his initial endeavors back as early as 1594
Before we move on from Napier, I do want to mention some interesting facts about him. Napier was so intelligent that the locals “believed him to be in league with the Devil”. Also the grass at his estate was greener (because he was gifted with agriculture) and he took late night walks in his nightgown and cap. In 1593, Napier published “A Plaine Discovery of the Whole Revelation of St. John” which described calculations he had done based on values found in the Book of Revelations that pointed to Pope Clement VIII being the “Antichrist”, and predicted that the end of the world would come in either 1688 or 1700. (Wright 2002)
The original logarithm introduced by both Napier and Burgi were very different from the logarithms we use today. Some of the modifications came soon after the original discovery. Others came later when exponential functions were discovered.
Henry Briggs was a professor at Gresham College in 1596. He was very interested in astronomy thus he did many calculations that involved multiplication of very large numbers. After reading Napier's work on logarithms, Briggs became very interested in the idea. He contacted Napier and together they made some very important adjustments.
In a letter sent just before their first meeting in 1615, Briggs suggested to Napier that logs should be in base 10 for convenience. Napier agreed with the idea, but could not take the time to make new tables due to poor health. During their meeting in 1615, Napier suggested that the new tables should also have the property that log 1 = 0.
Briggs compiled the tables as discussed, and published them in 1617, a year after Napier passed away.(O'Connor & Robertson, 1999)
Most of the time, logarithmic functions are introduce in the context of the inverse of exponential functions. Teaching it this manner leads to the common misconception that logarithms were founded for that very purpose. However, the concept of exponential functions date back to the later part of the seventeenth century while logarithms were officially discovered in 1614. The original logs, introduced by Napier, had no concept of “bases”. Natural logarithms based on the exponential function of base e appeared “almost contemporaneously with the Briggian logarithms (those logarithms motified by Henry Briggs in conjunction with Napier discussed above), but their fundamental importance was not recognized until the infinitesimal calculus was better understood.” (Struik, )