The properties of the logarithms, or the surprising ones ...

The need for computation appeared in a person immediately, as soon as he was able to give a quantitative assessment of the surrounding objects. It can be assumed that the logic of quantitative evaluation has gradually led to the need for "addition-subtraction" calculations. These two elementary actions are initially basic - all other manipulations with numbers, known as multiplication, division, exponentiation , etc. - this is a simple "mechanization" of some computational algorithms, based on the simplest arithmetic - "add-subtract". Whatever it was, but the creation of algorithms for computation is a major achievement of thought, and their authors forever leave their mark in the memory of mankind.

Six or seven centuries ago, in the field of marine navigation and astronomy, the need for large volumes of computation increased, which is not surprising, because It is the Middle Ages that is known for the development of navigation and astronomy. In exact accordance with the phrase "the need generates a sentence" of several mathematicians, the idea dawned - to replace the very laborious operation of multiplying two numbers by a simple addition (the idea of replacing division by subtraction was considered in a dual way). The working version of the new system of calculations was expounded in 1614 in the work of John Napier with the very remarkable title "Description of the amazing table of logarithms". Of course, further improvement of the new system continued, but the basic properties of the logarithms were laid out by Neper. The idea of a computation system using logarithms was that if a series of numbers forms a geometric progression, then their logarithms also form a progression, but an arithmetic one. In the presence of pre-compiled tables, a new calculation technique simplified the calculations, and the first logarithmic ruler (1620 ) became, perhaps, the first ancient and very effective calculator, an indispensable engineering tool.

For pioneers, the road is always bumpy. Initially, the base of the logarithm was taken unsuccessfully, and the accuracy of the calculations was not high, but already in 1624 the revised tables with decimal base were published. The properties of the logarithms follow from the essence of the definition: the logarithm of the number b is the number C, which, being the power of the base of the logarithm (the number A), results in the number b. The classical variant of the entry looks like this: logA (b) = C - what is read like this: logarithm b, on the base of A, is the number C. To perform actions using not quite ordinary logarithmic numbers, you need to know a certain set of rules known as "properties Logarithms ". In principle, all the rules have a common implication - how to add, subtract and transform logarithms. Now we will learn how to do it.

Logarithmic zero and unit

1. logA (1) = 0, the logarithm of the number 1 equals 0 for any reason - this is a direct consequence of raising the number to zero power.

2. logA (A) = 1, the logarithm of the same number with the base is 1 is also a 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 several numbers is equal to the logarithm of their product.

4. logA (m) - logA (n) = logA (m / n) - the difference between the logarithms of numbers, similarly to the previous one, is equal to the logarithm of the ratio of these numbers.

5. logA (1 / n) = - logA (n), the logarithm of the inverse number is equal to the logarithm of this number with a minus sign. It is easy to see that this is the result of the previous expression 4 for m = 1.

It is easy to see that rules 3-5 assume in both parts of equalities the same base of the logarithm.

The exponents in logarithmic expressions

6. logA (mn) = n * logA (m), the logarithm of a number of n is equal to the logarithm of this number multiplied by an exponent of degree n.

7. log (Ac) (b) = (1 / c) * logA (b), which is read as "the logarithm of the number b, if the base has the form Ac, is equal to the product of the logarithm b with the base A and the inverse of c".

The formula for changing the base of the logarithm

8. logA (b) = logC (b) / logc (A), the logarithm of the number b with base A on going to base C is calculated as the partial logarithm b with base C and the logarithm with base C of the number equal to the previous base A, and With a minus sign.

The above logarithms and their properties make it possible, with proper application, to simplify the calculation of large numeric arrays, thereby reducing the time of numerical calculations and providing acceptable accuracy.

It is not at all surprising that in science and technology the properties of the logarithms of numbers are used for a more natural representation of physical phenomena. For example, it is widely known to use the relative values - decibels for measuring the intensity of sound and light in physics, absolute stellar magnitude in astronomy, pH in chemistry, etc.

The efficiency of logarithmic calculations is easy to verify if one takes, for example, and multiplies 3 five-digit numbers "manually" (in a column), using tables of logarithms on a sheet of paper and using a logarithmic ruler. Suffice it to say that in the latter case, the calculations will take about 10 seconds. What is most surprising is that on a modern calculator these calculations will take no less time.