need for calculations appeared at the person right away, as soon as he was able to quantify the objects around him.We can assume that the logic of quantitative assessment gradually led to the need for a settlement of the "add-subtract".These two simple steps initially are the main - all the other manipulations of numbers known as multiplication, division, exponentiation, etc.- A simple, "mechanization" of some computational algorithms, which are based on simple arithmetic - "folded-subtract."Whatever it was, but the creation of algorithms for computing is a major achievement of thought, and their authors will forever leave its mark in the memory of mankind.
six or seven centuries ago in the field of maritime navigation and astronomy has increased the need for large amounts of computation, which is not surprising, sinceit is known to the Middle Ages, the development of navigation and astronomy.In keeping with the phrase "demand creates supply" several mathematicians had the idea - to replace a very time-consuming operation of multiplication of two numbers by simply adding (dually considered the idea to replace the division by subtraction).The working version of the new system of calculation was set out in 1614 in the work of John Napier's very remarkable title "Description of the table of logarithms wonderful."Of course, further improving the new system went on and on, but the basic properties of logarithms Napier has been presented.The idea of calculation using logarithms was the fact that if a series of numbers form a geometric progression, their logarithms also form a progression, but arithmetic.If you have a pre-compiled tables new method of doing calculations simplified the calculations, and the first slide rule (1620) was perhaps the first ancient and very effective calculator - an indispensable engineering tool.
for pioneering the road always with potholes.Initially, the base of the logarithm has been taken successfully and the accuracy of the calculations was low, but in 1624 were published refined table with a decimal base.The properties of logarithms are derived from the essence of the definition of the logarithm of b - is a number C, which, being the base of the logarithm of the degree (number A), resulting in a number of b.The classic version looks record: logA (b) = C - that read as follows: log b, the base A, is the number of C. To perform actions using the not quite normal, logarithmic number, you need to know a set of rules, known as "propertieslogarithms. "In principle, all rules have a common subtext - how to add, subtract and convert logarithms.Now we know how to do it.
logarithmic zero and one
1. logA (1) = 0, the logarithm of 1 is equal to 0 for any reason - is the direct result of a number raised to the zero power.
2. logA (A) = 1, the logarithm to the base of the same is 1 - also well-known truth for any number in the first degree.
Addition and subtraction of logarithms
3. logA (m) + logA (n) = logA (m * n) - the sum of the logarithms of numbers is equal to the logarithm of the number of their works.
4. logA (m) - logA (n) = logA (m / n) - the difference of logarithms, similar to the previous one, is equal to the logarithm of the ratio of these numbers.
5. logA (1 / n) = - logA (n), is equal to the logarithm of the inverse of the logarithm of this number with the sign "minus".It is easy to see that this is the result of the previous expression 4 with m = 1.
easy to see that the rules require 3-5 on both sides of the same base of the logarithm.
exponents in logarithmic terms
6. logA (mn) = n * logA (m), the logarithm of the number of degree n is the logarithm of the number of times the exponent n.
7. log (Ac) (b) = (1 / c) * logA (b), which reads like a "logarithm of b, if the base is given by Ac, is the product of the logarithm base b c A and the reciprocal c».
Formula changes logarithm base
8. logA (b) = - logC (b) / logc (A), the logarithm of b to the base A at the transition to the base C is calculated as the quotient of the logarithm with base b and C the logarithm to the basenumber equal to the previous base of A, and with the sign "minus".
listed above logarithms and their properties allow for a suitable application to simplify the calculation of the large numeric arrays, thereby reducing the time of the numerical calculations and provides acceptable accuracy.
It is not surprising that in science and engineering properties of logarithms are used to a more natural representation of physical phenomena.For example, is widely known to use relative values - decibels when measuring the intensity of sound and light in physics, the absolute magnitude of astronomy, in pH in chemistry and others.
Efficiency logarithmic computation is easy to check if you take, for example, and multiply 3 five-digit number"manually" (in a column), using tables of logarithms on a sheet of paper and the slide rule.Suffice it to say that in the latter case, the calculation will take on the strength of 10 seconds What is most surprising is the fact that in the modern calculator these calculations take time, not less.
