3 Answers

  1. Logarithms are needed to make it easier to compare values that differ several times. If you want to draw a graph or diagram in which a certain value can take the values of 10, 1000 and 1000000, then on a linear scale you will get that all the small values hang somewhere around zero, and on a logarithmic scale they can be depicted quite conveniently.

    When we evaluate the volume of sounds, we also have the ability to compare very different sounds in terms of power.

    Well, just logarithms naturally occur in some physical formulas.

  2. What is the logarithm? Look.. Let's say you have a certain number(a) and you raise it to the nth power and get a certain number(b). The logarithm has a base – this is the number(a) that you want to raise to some power to get the number b. For example, let's write it like this:

    log(2) 8 = 3, where in parentheses is the base of the logarithm. That is, this entry shows that the number(a), in this case – 2, must be raised to the power(N), in this case-3, to get the number(b) – 8

    For more information, see various literature sources, for example, see a school textbook for the 9th or 10th grade

  3. I will not write about the definition of logarithms, I will tell you why they are needed. Logarithms have many useful properties: first, the logarithm as a function grows very slowly (the natural logarithm is from 5 ≈ 0.7; from 5000 ≈ 3.7; from 5,000,000 ≈ 6.7), second, it grows monotonically [if a > b, then log(a) > > log(b)], and third, the logarithm of the product is equal to the sum of logarithms [log(a*b) = log(a)+log(b)]. All this allows, for example, to easily compare products of large quantities: you do not need to count the products themselves, just count their logarithms, and to find out the logarithm of the product, you can calculate the sum of logarithms from multipliers. In addition, logarithms are often useful for describing natural phenomena. For example, a person perceives some things “on a logarithmic scale”: the perceived difference between the volume of different sounds is not always the same, but is proportional to their volume (i.e., we hear the difference between two quiet sounds much better than between two loud ones). Because of this, perceived loudness levels (decibels) are calculated logarithmically. And in general, many things in nature, economics, and other fields are described using exponential laws (power laws). For example, the most frequent word in a large corpus of texts is usually found, roughly speaking, somewhere twice as often as the second in frequency, and it, in turn, is twice as often as the third in frequency, and so on (Zipf's law). Similar patterns are observed in the distribution of the size of cities and the strength of earthquakes. To visualize these dimensions and work with them, it is convenient to use their logarithms rather than real values.

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