Convenient formulas are ones in which the operations that depend on logarithms are done all at once. The recourse to the tables then consists of only two steps. One is obtaining logarithms, the other obtaining antilogs.
The procedures of trigonometry were recast to produce such formulas. Logarithm , mathematical power or exponent to which any particular number, called the base, is raised in order to produce another particular number. Common logarithms use the number 10 as the base. Natural logarithms use the transcendental number e as a base. The first tables of logarithms were published independently by Scottish mathematician John Napier in and Swiss mathematician Justus Byrgius in The problem in constructing a table of logarithms is to make the intervals between successive entries small enough for usefulness in calculating.
Logarithm tables have been replaced by electronic calculators and computers with logarithmic functions. Each logarithm contains a whole number and a decimal fraction, called respectively the characteristic and the mantissa. In the common system of logarithms, the logarithm of the number 7 has the characteristic 0 and the mantissa.
The logarithm of the number 70 is 1. The logarithm of the number. Last Updated on 2 July, For suggestions please mail the editor in chief. John Napier1 was born in Edinburgh Scotland. 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.
Napier replied that he too had the idea but could not create the tables due to an illness. Summer Henry Briggs visited John Napier and they spent a month working on the tables for the logarithms to base Henry Briggs visited John Napier a second time. This is one of the tricks that Napier used. Robert Flower claimed that he could compute a logarithm to 20 decimal places in 7 or 8 minutes, with nothing more than a pencil and paper.
I wondered if there could be a way to compute logarithms on an abacus. I later found out that it is not possible. You need two of them. What about a recursive method that converges quadratically? You could get ten digit logs from your calculator, do one iteration, and have them to twenty digits. Yep, there is such an algorithm. John Napier published his table of logarithms Mirifici Logarithmorum Canonis Descriptio in after some twenty years of work and described his method of construction in Mirifici Logarithmorum Canonis Constructio , published posthumously in Edinburgh by his son Robert, with appendices by Napier and Henry Briggs Briggs worked with Napier on improving the methods of calculation in the summers following the publication of the table until Napier's death and published his own method in Vlacq's edition.
The following is based on the Constructio , Macdonald's translation , and Goldstine's summary in A History of Numerical Analysis from the 16th through the 19th Century , Properties of geometric and arithmetic progressions were well-known by Napier's time, and the connection between a sequence of powers and its corresponding sequence of exponents that we call the law of exponents has roots in ancient mathematics cf.
Euclid IX. I do not know what enabled Napier to make the key connection that led to logarithms or on the other hand, what prevented the discovery earlier. My take on Napier is that calculating was thought primarily in terms of integers or ratios of integers or fractions , and Napier is at pains to make his calculation of logarithms in terms of integers accurate.
His break-through idea is quite ingenious and surprising to me, at least. It is to consider two points moving continuously, one representing the logarithm increasing "arithmetically" while the other representing what we would now call the argument decreasing "geometrically. Like many first tries, this led to some inconveniences. By the time his table was published, Napier had in mind some improvements, which are included in an appendix to the Constructio.
It is large, it appears, so that one can express logarithms accurately in terms of integers. Briggs too saw some room for improvement, and this led him to seek out Napier. By default, I pick the natural logarithm. Radians are similar: they measure angles in terms of the mover. Not yet. They give us a rate as if all the change happened in a single time period.
The change could indeed be a single year of A banker probably cares about the human-friendly, year-over-year difference. We can figure this out by letting the continuous growth run for a year:. The year-over-year gain is 3. From an instant-by-instant basis, a given part of the economy is growing by 3. In science and engineering, we prefer modeling behavior on an instantaneous basis. Do bacteria colonies replicate on clean human intervals, and do we wait around for an exact doubling?
We know the rate was. Figuring out whether you want the input cause of growth or output result of growth is pretty straightforward. Imagine we have little workers who are building the final growth pattern see the article on exponents :. Green at the end of the year. But… that worker he was building Mr. Green starts working as well. If Mr. Green first appears at the 6-month mark, he has a half-year to work same annual rate as Mr. Blue and he builds Mr.
Of course, Mr. Red ends up being half done, since Mr. Green only has 6 months. What if Mr. Green showed up after 4 months? A month? A day? A second? If workers begin growing immediately, we get the instant-by-instant curve defined by ex:. We plug that rate into ex to find the final result, with all compounding included. Using Other Bases Switching to another type of logarithm base 10, base 2, etc. Log base e: What was the instantaneous rate followed by each worker?
Log base 2: How many doublings were required? Log base How many 10x-ings were required? Over 30 years, the transistor counts on typical chips went from to 1 billion How would you analyze this?
Doubling is easier to think about than 10x-ing. With these assumptions we get:. Summary Learning is about finding the hidden captions behind a concept. When is it used? What point view does it bring to the problem?
My current interpretation is that exponents ask about cause vs. In the real world, calculators may lose precision, so use a direct log base 2 function if possible. I a non-mathematician fan of the subject was suddenly seized with a desire to know how e was implicit in Napier's construction.
Your article is the most lucid explanation I could find. Thanks heaps! Skip to main content. Marianne Freiberger. Greek astronomer. In the 19th century A. Most other logarithmic scales have a similar story. That logarithmic scales often come first suggests that they are, in a sense, intuitive. This not only has to do with our perception, but also how we instinctively think about numbers. Though logarithmic scales are troublesome to many if not most math students, they strangely have a lot to do with how we all instinctively thought about numbers as infants.
A change from eight ducks to 16 ducks caused activity in the parietal lobe, showing that newborns have an intuition of numbers. When thinking in terms of ratios and logarithmic scales rather than differences and linear scales , one times three is three, and three times three is nine, so three is in the middle of one and nine. These were of particular utility for simplifying calculations.
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