The need for calculations appeared in a person immediately, as soon as he managed to quantify the objects surrounding him. It can be assumed that the logic of quantitative assessment gradually led to the need for addition-subtraction calculations. These two simplest actions are initially basic - all other manipulations with numbers, known as multiplication, division, exponentiation , etc. - This is a simple “mechanization” of certain computational algorithms, which are based on the simplest arithmetic - “add-subtract”. Be that as it may, the creation of calculation algorithms is a major achievement of thought, and their authors will leave their mark in the memory of mankind forever.
Six to seven centuries ago, in the field of marine navigation and astronomy, the need for large amounts of computing has increased, which is not surprising, because it is the Middle Ages that are known for the development of navigation and astronomy. In exact accordance with the phrase “need gives rise to a proposal”, several mathematicians came up with the idea of ​​replacing the very laborious operation of multiplying two numbers with simple addition (the idea of ​​replacing division by subtraction was also considered dually). A working version of the new computing system was presented in 1614 in the work of John Napier with a very remarkable title "Description of the amazing table of logarithms." Of course, further improvement of the new system continued further, but the main properties of the logarithms were stated by Naper. The idea of ​​a calculation system using logarithms was that if a certain series of numbers forms a geometric progression, then their logarithms also form a progression, but already arithmetic. If you have pre-compiled tables, a new calculation method simplified the calculations, and the first slide rule (1620 ) was perhaps the first ancient and very effective calculator - an indispensable engineering tool.
For pioneers, the road is always with bumps. Initially, the base of the logarithm was taken unsuccessfully, and the accuracy of the calculations was not high, but already in 1624, updated tables with a decimal base were published. The properties of the logarithms follow from the essence of the definition: the logarithm of the number b is a number C which, being the degree of the base of the logarithm (number A), results in the number b. The classic version of the record looks like this: logA (b) = C - which is read like this: the logarithm of b, on the basis 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 subtext - how to add, subtract and convert logarithms. Now we will find out how to do it.
Logarithmic zero and one
1. logA (1) = 0, the logarithm of the number 1 is 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 as the base is 1 - also a well-known truth for any number in the first degree.
Logarithm Addition and Subtraction
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, similar 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 with m = 1.
It is easy to see that rules 3-5 assume the same base of the logarithm in both parts of the equalities.
Exponents in logarithmic expressions
6. logA (mn) = n * logA (m), the logarithm of a number in degree n is equal to the logarithm of this number times the exponent n.
7. log (Ac) (b) = (1 / c) * logA (b), which reads as “the logarithm of b if the base is of the form Ac is equal to the product of the logarithm of b with base A and the number 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 when passing to base C is calculated as the quotient of the logarithm of b with base C and the logarithm of base C with the number equal to the previous base A, and with a minus sign.
The logarithms listed above and their properties make it possible to simplify the calculation of large numerical arrays with proper use, which reduces the time for numerical calculations and ensures acceptable accuracy.
It is not at all surprising that in science and technology, the properties of the logarithms of numbers are used to more naturally represent physical phenomena. For example, it is widely known the use of relative values ​​- decibels in measuring the intensity of sound and light in physics, absolute magnitude in astronomy, pH in chemistry, etc.
The effectiveness of logarithmic calculations can be easily checked by taking, for example, and multiplying 3 five-digit numbers “manually” (in a column), using the tables of logarithms on a sheet of paper and using a slide rule. It is enough 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.