With the development of society, the complexity of production, mathematics also developed. Movement from simple to complex. From the usual accounting method of addition and subtraction, with their repeated repetition, they came to the concept of multiplication and division. The reduction of the multiply repeated operation became the concept of exponentiation. The first tables of the dependence of numbers on the base and the number of exponentiation were compiled back in the 8th century by the Indian mathematician Varasena. From them, you can count the time of occurrence of logarithms.

Historical outline

The revival of Europe in the 16th century also stimulated the development of mechanics. T required a large amount of computation associated with multiplication and division of multi-digit numbers. The ancient tables did a great service. They made it possible to replace complex operations with simpler ones - addition and subtraction. A big step forward was the work of the mathematician Michael Stiefel, published in 1544, in which he realized the idea of ​​many mathematicians. This made it possible to use tables not only for degrees in the form of prime numbers, but also for arbitrary rational ones.

In 1614, the Scotsman John Napier, developing these ideas, first introduced the new term "logarithm of a number." New complex tables were compiled for calculating the logarithms of sines and cosines, as well as tangents. This greatly reduced the work of astronomers.

New tables began to appear, which were successfully used by scientists for three centuries. A lot of time passed before the new operation in algebra acquired its finished form. The logarithm was defined and its properties were studied.

Only in the 20th century, with the advent of the calculator and the computer, mankind abandoned the ancient tables that had been successfully operating throughout the 13th centuries.

Today we call the logarithm of b to base a the number x, which is the power of a, to get the number b. This is written as a formula: x = log a(b).

For example, log 3(9) will be equal to 2. This is obvious if you follow the definition. If we raise 3 to the power of 2, we get 9.

Thus, the formulated definition puts only one restriction, the numbers a and b must be real.

Varieties of logarithms

The classical definition is called the real logarithm and is actually a solution to the equation a x = b. The option a = 1 is borderline and is of no interest. Note: 1 to any power is 1.

Real value of the logarithm defined only if the base and the argument is greater than 0, and the base must not be equal to 1.

Special place in the field of mathematics play logarithms, which will be named depending on the value of their base:

Rules and restrictions

The fundamental property of logarithms is the rule: the logarithm of a product is equal to the logarithmic sum. log abp = log a(b) + log a(p).

As a variant of this statement, it will be: log c (b / p) \u003d log c (b) - log c (p), the quotient function is equal to the difference of the functions.

It is easy to see from the previous two rules that: log a(b p) = p * log a(b).

Other properties include:

Comment. Do not make a common mistake - the logarithm of the sum is not equal to the sum of the logarithms.

For many centuries, the operation of finding the logarithm was a rather time-consuming task. Mathematicians used the well-known formula of the logarithmic theory of expansion into a polynomial:

ln (1 + x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 + ... + ((-1)^(n + 1))*(( x^n)/n), where n is a natural number greater than 1, which determines the accuracy of the calculation.

Logarithms with other bases were calculated using the theorem on the transition from one base to another and the property of the logarithm of the product.

Since this method is very laborious and when solving practical problems difficult to implement, they used pre-compiled tables of logarithms, which greatly accelerated the entire work.

In some cases, specially compiled graphs of logarithms were used, which gave less accuracy, but significantly speeded up the search for the desired value. The curve of the function y = log a(x), built on several points, allows using the usual ruler to find the values ​​of the function at any other point. For a long time, engineers used the so-called graph paper for these purposes.

In the 17th century, the first auxiliary analog computing conditions appeared, which by the 19th century had acquired a finished form. The most successful device was called the slide rule. Despite the simplicity of the device, its appearance significantly accelerated the process of all engineering calculations, and this is difficult to overestimate. Currently, few people are familiar with this device.

The advent of calculators and computers made it pointless to use any other devices.

Equations and inequalities

The following formulas are used to solve various equations and inequalities using logarithms:

• Transition from one base to another: log a(b) = log c(b) / log c(a);
• As a consequence of the previous version: log a(b) = 1 / log b(a).

To solve inequalities, it is useful to know:

• The value of the logarithm will only be positive if both the base and the argument are both greater than or less than one; if at least one condition is violated, the value of the logarithm will be negative.
• If the logarithm function is applied to the right and left sides of the inequality, and the base of the logarithm is greater than one, then the sign of the inequality is preserved; otherwise, it changes.

Task examples

Consider several options for using logarithms and their properties. Examples with solving equations:

Consider the option of placing the logarithm in the degree:

• Task 3. Calculate 25^log 5(3). Solution: in the conditions of the problem, the notation is similar to the following (5^2)^log5(3) or 5^(2 * log 5(3)). Let's write it differently: 5^log 5(3*2), or the square of a number as a function argument can be written as the square of the function itself (5^log 5(3))^2. Using the properties of logarithms, this expression is 3^2. Answer: as a result of the calculation we get 9.

Practical use

Being a purely mathematical tool, it seems far removed from real life that the logarithm has suddenly become of great importance in describing objects in the real world. It is difficult to find a science where it is not used. This fully applies not only to the natural, but also to the humanities fields of knowledge.

Logarithmic dependencies

Here are some examples of numerical dependencies:

Mechanics and physics

Historically, mechanics and physics have always developed using mathematical research methods and at the same time served as an incentive for the development of mathematics, including logarithms. The theory of most laws of physics is written in the language of mathematics. We give only two examples of the description of physical laws using the logarithm.

It is possible to solve the problem of calculating such a complex quantity as the speed of a rocket using the Tsiolkovsky formula, which laid the foundation for the theory of space exploration:

V = I * ln(M1/M2), where

• V is the final speed of the aircraft.
• I is the specific impulse of the engine.
• M 1 is the initial mass of the rocket.
• M 2 - final mass.

Another important example- this is the use in the formula of another great scientist, Max Planck, which serves to evaluate the equilibrium state in thermodynamics.

S = k * ln (Ω), where

• S is a thermodynamic property.
• k is the Boltzmann constant.
• Ω is the statistical weight of different states.

Chemistry

Less obvious would be the use of formulas in chemistry containing the ratio of logarithms. Here are just two examples:

• The Nernst equation, the condition of the redox potential of the medium in relation to the activity of substances and the equilibrium constant.
• The calculation of such constants as the autoprolysis index and the acidity of the solution is also not complete without our function.

Psychology and biology

And it’s completely incomprehensible what the psychology has to do with it. It turns out that the strength of sensation is well described by this function as the inverse ratio of the stimulus intensity value to the lower intensity value.

After the above examples, it is no longer surprising that the theme of logarithms is also widely used in biology. Whole volumes can be written about biological forms corresponding to logarithmic spirals.

Other areas

It seems that the existence of the world is impossible without connection with this function, and it governs all laws. Especially when the laws of nature are connected with a geometric progression. It is worth referring to the MatProfi website, and there are many such examples in the following areas of activity:

The list could be endless. Having mastered the basic laws of this function, you can plunge into the world of infinite wisdom.

Power or logarithmic dependence?

Comparison of correlation coefficients

Back in the 19th century German philosopher, one of the founders of scientific psychology G.-T. Fechner put forward a psychophysical law describing the dependence of sensations on the magnitude of physical stimulation. This law, called the Weber-Fechner law, assumed a logarithmic relationship between the energy of the stimulus acting on the sense organ and the magnitude of the sensation that this stimulus causes. In the XX century. the American psychophysicist S. S. Stevens criticized Fechner's methodology, which did not imply the possibility of a direct assessment of sensation. The result of this criticism was the development by S. S. Stevens of a number of methodological procedures, which were called methods of direct assessment of sensations. Based on the data obtained in the experiment, it became possible to evaluate the relationship between the magnitude of the stimulus and the magnitude of sensation not only in theory, but also in practice. As a result, Stevens concluded that psychophysical dependence should be described but logarithmic, a power function.

Let's see how the Stevens methodology and the simplest procedures of correlation analysis make it possible to compare the data for their compliance with the logarithmic and power law psychophysical.

To do this, we will use the results obtained in one psychophysical experiment (T. Engen). In this experiment, the modulus value method was used to estimate the odor concentrations of amyl acetate (banana) diluted in diethyl phthalate. Each of the 12 subjects evaluated seven different odor concentrations twice. A concentration of 12.5% ​​was used as the modulus. The modulus value was set equal to 10. 7.10 presents the average scale values ​​for each stimulus.

We present these results in the form of a scatterplot (Fig. 7.7). It can be seen that as the concentration of an odorous substance increases, the subjective assessment of its sensation increases. This dependence is monotonic, but apparently non-linear. However, calculating the correlation coefficient between these two data series gives a rather high value of 0.984. Such a correlation coefficient explains 96.8% of the variance of the dependent variable (criterion) directly associated with the value of the independent variable (predictor), although it does not have any theoretical basis.

Table 7.10

Subjective odor scale of amyl acetate diluted in diatyl phthalate (T. Engen )

Rice. 7.7.

The logarithmic Weber–Fechner law suggests that a linear relationship will be observed between the logarithms of amyl acetate concentration and the subjective sensation score.

Such a dependence seems very likely, judging by the data presented in Fig. 7.7. Therefore, we will transform the concentrations used in the experiment into their natural logarithms and again construct a scatterplot. On fig. 7.8 reflects the dependence of the subjective assessment of the smell, now on the value of the logarithm of the concentration of amyl acetate. But again, as it seems, we do not observe a linear relationship. This time, the correlation coefficient between the logarithm of the concentration of an odorous substance and the subjective assessment of its odor turned out to be even lower than what we noted for the original data, although still quite high - 0.948. In this case, only 89.8% of the test variance is directly related to the predictor variance. Thus, the predictions of the Weber-Fechner law in relation to our data do not look very convincing.

Rice. 7.8.

The power-law Stevens psychophysical law establishes a linear relationship between the logarithms of stimulation and the magnitude of sensation. Figure 7.9 shows that this prediction is quite accurate. All points of the scatterplot line up perfectly along one line. The correlation coefficient between these data series is 0.999. This means that such a regression model describes 99.8% of the variance of the dependent variable that can be related to the variance of the independent variable.

Rice. 7.9.

Thus, a visual comparison of Fig. 7.7-7.9, as well as the calculated correlation coefficients, seem to unequivocally testify in favor of the Stevens power law. Nevertheless, let's try to estimate how big the statistical difference between these three correlation coefficients is.

First of all, we will carry out a logarithmic transformation of the correlation coefficients calculated by us, using the non-linear Fisher transform:

To simplify the calculations, you can use the corresponding function Microsoft Excel - FISHER. As an argument, it takes the value of the corresponding correlation coefficient.

The results of such transformations give us the following values ​​of z":

• 1. For the relationship between amyl acetate concentrations and odor assessment, z" = 2.41.
• 2. For the connection between the logarithm of concentrations and the assessment of odors, z" = 1.81.
• 3. For the connection between the logarithm of concentrations and the logarithm of subjective estimates, z" = 3.89.

Now we can put forward three statistical hypotheses about the pairwise equality of these correlation coefficients in the general population. To assess the statistical reliability of these hypotheses, it is necessary to construct three statistics z :

Here P and t match the sample sizes. In our case, both values ​​are equal to seven, since the same data are used.

As a result, we get that the statistics z for the case of comparing the correlation coefficient between the initial values ​​of the concentration of an odorous substance and the subjective assessment of the smell, on the one hand, and the correlation coefficient between the results of the logarithmic transformation of stimulus values ​​and their sensations, on the other hand, it turns out to be equal to 0.85, which corresponds to the Weber-Fechner law. The reliability of these statistics can be assessed using statistical tables (see Appendix 1). The estimate shows that such a value is not reliably different from zero and, therefore, it is necessary to maintain the null hypothesis put forward about the equality of these correlation coefficients.

Comparison of the correlation coefficient, which assumes the logarithmic transformation of both variables - Stevens' law, with the correlation coefficients, which assumes the logarithmic transformation of only the independent variable - the Weber-Fechner law and does not imply such a transformation at all, gives z-statistic values ​​of 2.94 and 2.10, respectively. Both of these values ​​indicate a reliable difference between the z statistics and the theoretically expected zero value. Consequently,

it is necessary to reject the null hypothesis about the equality of the correlation coefficients.

(from the Greek λόγος - "word", "relation" and ἀριθμός - "number") numbers b by reason a(log α b) is called such a number c, and b= a c, that is, log α b=c and b=ac are equivalent. The logarithm makes sense if a > 0, a ≠ 1, b > 0.

In other words logarithm numbers b by reason a formulated as an exponent to which a number must be raised a to get the number b(the logarithm exists only for positive numbers).

From this formulation it follows that the calculation x= log α b, is equivalent to solving the equation a x =b.

For example:

log 2 8 = 3 because 8=2 3 .

We note that the indicated formulation of the logarithm makes it possible to immediately determine logarithm value when the number under the sign of the logarithm is a certain power of the base. Indeed, the formulation of the logarithm makes it possible to justify that if b=a c, then the logarithm of the number b by reason a equals With. It is also clear that the topic of logarithm is closely related to the topic degree of number.

The calculation of the logarithm is referred to logarithm. Logarithm is the mathematical operation of taking a logarithm. When taking a logarithm, the products of factors are transformed into sums of terms.

Potentiation is the mathematical operation inverse to logarithm. When potentiating, the given base is raised to the power of the expression on which the potentiation is performed. In this case, the sums of terms are transformed into the product of factors.

Quite often, real logarithms with bases 2 (binary), e Euler number e ≈ 2.718 (natural logarithm) and 10 (decimal) are used.

At this stage, it is worth considering samples of logarithms log 7 2 , ln √ 5, lg0.0001.

And the entries lg (-3), log -3 3.2, log -1 -4.3 do not make sense, since in the first of them a negative number is placed under the sign of the logarithm, in the second - a negative number in the base, and in the third - and a negative number under the sign of the logarithm and unit in the base.

Conditions for determining the logarithm.

It is worth considering separately the conditions a > 0, a ≠ 1, b > 0. definition of a logarithm. Let's consider why these restrictions are taken. This will help us with an equality of the form x = log α b, called the basic logarithmic identity, which directly follows from the definition of the logarithm given above.

Take the condition a≠1. Since one is equal to one to any power, then the equality x=log α b can only exist when b=1, but log 1 1 will be any real number. To eliminate this ambiguity, we take a≠1.

Let us prove the necessity of the condition a>0. At a=0 according to the formulation of the logarithm, can only exist when b=0. And then accordingly log 0 0 can be any non-zero real number, since zero to any non-zero power is zero. To eliminate this ambiguity, the condition a≠0. And when a<0 we would have to reject the analysis of the rational and irrational values ​​of the logarithm, since the exponent with a rational and irrational exponent is defined only for non-negative bases. It is for this reason that the condition a>0.

And the last condition b>0 follows from the inequality a>0, because x=log α b, and the value of the degree with a positive base a always positive.

Features of logarithms.

Logarithms characterized by distinctive features, which led to their widespread use to greatly facilitate painstaking calculations. In the transition "to the world of logarithms", multiplication is transformed into a much easier addition, division into subtraction, and raising to a power and taking a root are transformed into multiplication and division by an exponent, respectively.

The formulation of logarithms and a table of their values ​​(for trigonometric functions) was first published in 1614 by the Scottish mathematician John Napier. Logarithmic tables, enlarged and detailed by other scientists, were widely used in scientific and engineering calculations, and remained relevant until electronic calculators and computers began to be used.

logarithmic dependence- logaritminė priklausomybė statusas T sritis fizika atitikmenys: engl. logarithmic dependence vok. logarithmische Abhängigkeit, f rus. logarithmic dependence, fpranc. dependance logarithmique, f … Fizikos terminų žodynas

The function inverse to the exponential function (See exponential function). L. f. denoted y = lnx; (1) its value y, corresponding to the value of the argument x, is called the natural logarithm of the number x. By definition...

Paper cut in a special way; usually printed. It is constructed as follows (Fig. 1): on each of the axes of a rectangular coordinate system, decimal logarithms of numbers u are plotted (on the x-axis) and ... Great Soviet Encyclopedia

The function inverse to the exponential function. L. f. its value y is denoted, corresponding to the value of the argument x, called. the natural logarithm of x. By definition, relation (1) is equivalent Since for any real y, then L. f. ... ... Mathematical Encyclopedia

Graph of the binary logarithm Logarithm of a number ... Wikipedia

Weber-Fechner law- logarithmic dependence of the strength of sensation E on the physical intensity of the stimulus P: E = k log P + c, where k and c are some constants determined by this sensory system. The dependence was derived by the German psychologist and physiologist G. T. Fechner ...

sensation intensity- the degree of subjective severity of sensation associated with a certain stimulus. The relationship between the intensity of sensation and the physical intensity of the stimulus is quite complex. Various models have been proposed to describe this relationship: for example, in ... ... Great Psychological Encyclopedia

Weber-Fechner law- logarithmic dependence of the strength of sensation (E) on the physical intensity of the stimulus (P): E \u003d k log P + + c, where k and c are some constants determined by this sensory system. This dependence was derived by the German psychologist and physiologist G. T ... Great Psychological Encyclopedia

I. Task P.; II. laws of Weber and Fechner; III. Psychophysical methods; IV. Experimental results; V. Meaning of psychophysical laws; VI. Literature. I. Task P. Comparing different sensations, we notice that they have: 1) different qualities, 2) ... ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron

The flow of a liquid or gas, characterized by chaotic, irregular movement of its volumes and their intense mixing (see Turbulence), but in general, having a smooth, regular character. The formation of T. t. is associated with instability ... ... Encyclopedia of technology

basic psychophysical law- THE BASIC PSYCHO-PHYSICAL LAW - the function of the dependence of the magnitude of sensation on the magnitude of the stimulus. A single formula O. p. z. no, but there are variants of it: logarithmic (Fechner), power (Stevens), generalized (Baird, Ekman, Zabrodin, etc.) ... Encyclopedia of Epistemology and Philosophy of Science

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