The focus of this article is logarithm. Here we will give the definition of the logarithm, show the accepted notation, give examples of logarithms, and talk about natural and decimal logarithms. After that, consider the basic logarithmic identity.
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Definition of logarithm
The concept of a logarithm arises when solving a problem in a certain sense inverse, when you need to find the exponent from a known value of the degree and a known base.
But enough preamble, it's time to answer the question "what is a logarithm"? Let us give an appropriate definition.
Definition.
Logarithm of b to base a, where a>0 , a≠1 and b>0 is the exponent to which you need to raise the number a to get b as a result.
At this stage, we note that the spoken word "logarithm" should immediately raise two ensuing questions: "what number" and "on what basis." In other words, there is simply no logarithm, but there is only the logarithm of a number in some base.
We will immediately introduce logarithm notation: the logarithm of the number b to the base a is usually denoted as log a b . The logarithm of the number b to the base e and the logarithm to the base 10 have their own special designations lnb and lgb respectively, that is, they write not log e b , but lnb , and not log 10 b , but lgb .
Now you can bring: .
And the records 
do not make sense, since in the first of them there is a negative number under the sign of the logarithm, in the second - a negative number in the base, and in the third - both a negative number under the sign of the logarithm and a unit in the base.
Now let's talk about rules for reading logarithms. The entry log a b is read as "logarithm of b to base a". For example, log 2 3 is the logarithm of three to base 2, and is the logarithm of two integer two base thirds of the square root of five. The logarithm to base e is called natural logarithm, and the notation lnb is read as "the natural logarithm of b". For example, ln7 is the natural logarithm of seven, and we will read it as the natural logarithm of pi. The logarithm to base 10 also has a special name - decimal logarithm, and the notation lgb is read as "decimal logarithm b". For example, lg1 is the decimal logarithm of one, and lg2.75 is the decimal logarithm of two point seventy-five hundredths.
It is worth dwelling separately on the conditions a>0, a≠1 and b>0, under which the definition of the logarithm is given. Let us explain where these restrictions come from. To do this, we will be helped by an equality of the form, called , which directly follows from the definition of the logarithm given above.
Let's start with a≠1 . Since one is equal to one to any power, then the equality can only be true for b=1, but log 1 1 can be any real number. To avoid this ambiguity, a≠1 is accepted.
Let us substantiate the expediency of the condition a>0 . With a=0, by the definition of the logarithm, we would have equality , which is possible only with b=0 . But then log 0 0 can be any non-zero real number, since zero to any non-zero power is zero. This ambiguity can be avoided by the condition a≠0 . And for a<0 нам бы пришлось отказаться от рассмотрения рациональных и иррациональных значений логарифма, так как степень с рациональным и иррациональным показателем определена лишь для неотрицательных оснований. Поэтому и принимается условие a>0 .
Finally, the condition b>0 follows from the inequality a>0 , since , and the value of the degree with a positive base a is always positive.
In conclusion of this paragraph, we say that the voiced definition of the logarithm allows you to immediately indicate the value of the logarithm when the number under the sign of the logarithm is a certain degree of base. Indeed, the definition of the logarithm allows us to assert that if b=a p , then the logarithm of the number b to the base a is equal to p . That is, the equality log a a p =p is true. For example, we know that 2 3 =8 , then log 2 8=3 . We will talk more about this in the article.
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 it is difficult to overestimate this. 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.
Let's start with properties of the logarithm of unity. Its formulation is as follows: the logarithm of unity is equal to zero, that is, log a 1=0 for any a>0 , a≠1 . The proof is straightforward: since a 0 =1 for any a that satisfies the above conditions a>0 and a≠1 , then the proven equality log a 1=0 immediately follows from the definition of the logarithm.
Let's give examples of application of the considered property: log 3 1=0 , lg1=0 and .
Let's move on to the next property: the logarithm of a number equal to the base is equal to one, i.e, log a a=1 for a>0 , a≠1 . Indeed, since a 1 =a for any a , then by the definition of the logarithm log a a=1 .
Examples of using this property of logarithms are log 5 5=1 , log 5.6 5.6 and lne=1 .
For example, log 2 2 7 =7 , log10 -4 =-4 and 
.
Logarithm of the product of two positive numbers x and y is equal to the product of the logarithms of these numbers: log a (x y)=log a x+log a y, a>0 , a≠1 . Let us prove the property of the logarithm of the product. Due to the properties of the degree a log a x+log a y =a log a x a log a y, and since by the main logarithmic identity a log a x =x and a log a y =y , then a log a x a log a y =x y . Thus, a log a x+log a y =x y , whence the required equality follows by the definition of the logarithm.
Let's show examples of using the property of the logarithm of the product: log 5 (2 3)=log 5 2+log 5 3 and 
.
The product logarithm property can be generalized to the product of a finite number n of positive numbers x 1 , x 2 , …, x n as log a (x 1 x 2 ... x n)= log a x 1 + log a x 2 +…+ log a x n . This equality is easily proved.
For example, the natural logarithm of a product can be replaced by the sum of three natural logarithms of the numbers 4 , e , and .
Logarithm of the quotient of two positive numbers x and y is equal to the difference between the logarithms of these numbers. The quotient logarithm property corresponds to a formula of the form , where a>0 , a≠1 , x and y are some positive numbers. The validity of this formula is proved like the formula for the logarithm of the product: since 
, then by the definition of the logarithm .
Here is an example of using this property of the logarithm: 
.
Let's move on to property of the logarithm of degree. The logarithm of a degree is equal to the product of the exponent and the logarithm of the modulus of the base of this degree. We write this property of the logarithm of the degree in the form of a formula: log a b p =p log a |b|, where a>0 , a≠1 , b and p are numbers such that the degree of b p makes sense and b p >0 .
We first prove this property for positive b . The basic logarithmic identity allows us to represent the number b as a log a b , then b p =(a log a b) p , and the resulting expression, due to the power property, is equal to a p log a b . So we come to the equality b p =a p log a b , from which, by the definition of the logarithm, we conclude that log a b p =p log a b .
It remains to prove this property for negative b . Here we note that the expression log a b p for negative b makes sense only for even exponents p (since the value of the degree b p must be greater than zero, otherwise the logarithm will not make sense), and in this case b p =|b| p . Then b p =|b| p =(a log a |b|) p =a p log a |b|, whence log a b p =p log a |b| .
For example, 
and ln(-3) 4 =4 ln|-3|=4 ln3 .
It follows from the previous property property of the logarithm from the root: the logarithm of the root of the nth degree is equal to the product of the fraction 1/n and the logarithm of the root expression, that is, 
, where a>0 , a≠1 , n is a natural number greater than one, b>0 .
The proof is based on the equality (see ), which is valid for any positive b , and the property of the logarithm of the degree: 
.
Here is an example of using this property: 
.
Now let's prove conversion formula to the new base of the logarithm kind 
. To do this, it suffices to prove the validity of the equality log c b=log a b log c a . The basic logarithmic identity allows us to represent the number b as a log a b , then log c b=log c a log a b . It remains to use the property of the logarithm of the degree: log c a log a b = log a b log c a. Thus, the equality log c b=log a b log c a is proved, which means that the formula for the transition to a new base of the logarithm is also proved.
Let's show a couple of examples of applying this property of logarithms: and 
.
The formula for moving to a new base allows you to move on to working with logarithms that have a “convenient” base. For example, it can be used to go to natural or decimal logarithms so that you can calculate the value of the logarithm from the table of logarithms. The formula for the transition to a new base of the logarithm also allows in some cases to find the value of a given logarithm, when the values of some logarithms with other bases are known.
Often used is a special case of the formula for the transition to a new base of the logarithm for c=b of the form 
. This shows that log a b and log b a – . For example, 
.
Also often used is the formula 
, which is useful for finding logarithm values. To confirm our words, we will show how the value of the logarithm of the form is calculated using it. We have 
. To prove the formula 
it is enough to use the transition formula to the new base of the logarithm a: 
.
It remains to prove the comparison properties of logarithms.
Let us prove that for any positive numbers b 1 and b 2 , b 1 log a b 2 , and for a>1, the inequality log a b 1 Finally, it remains to prove the last of the listed properties of logarithms. We confine ourselves to proving its first part, that is, we prove that if a 1 >1 , a 2 >1 and a 1 1 is true log a 1 b>log a 2 b . The remaining statements of this property of logarithms are proved by a similar principle. Let's use the opposite method. Suppose that for a 1 >1 , a 2 >1 and a 1 1 log a 1 b≤log a 2 b is true. By the properties of logarithms, these inequalities can be rewritten as 
and 
respectively, and from them it follows that log b a 1 ≤log b a 2 and log b a 1 ≥log b a 2, respectively. Then, by the properties of powers with the same bases, the equalities b log b a 1 ≥b log b a 2 and b log b a 1 ≥b log b a 2 must be satisfied, that is, a 1 ≥a 2 . Thus, we have arrived at a contradiction to the condition a 1
Bibliography.
- Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the Beginnings of Analysis: A Textbook for Grades 10-11 of General Educational Institutions.
- Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).
In relation to
the task of finding any of the three numbers from the other two, given, can be set. Given a and then N is found by exponentiation. If N are given and then a is found by extracting the root of the power x (or exponentiation). Now consider the case when, given a and N, it is required to find x.
Let the number N be positive: the number a is positive and not equal to one: .
Definition. The logarithm of the number N to the base a is the exponent to which you need to raise a to get the number N; the logarithm is denoted by
Thus, in equality (26.1), the exponent is found as the logarithm of N to the base a. Entries
have the same meaning. Equality (26.1) is sometimes called the basic identity of the theory of logarithms; in fact, it expresses the definition of the concept of the logarithm. By this definition, the base of the logarithm a is always positive and different from unity; the logarithmable number N is positive. Negative numbers and zero do not have logarithms. It can be proved that any number with a given base has a well-defined logarithm. Therefore equality entails . Note that the condition is essential here, otherwise the conclusion would not be justified, since the equality is true for any values of x and y.
Example 1. Find
Decision. To get the number, you need to raise base 2 to the power Therefore.
You can record when solving such examples in the following form:
Example 2. Find .
Decision. We have
In examples 1 and 2, we easily found the desired logarithm by representing the logarithmable number as a degree of base with a rational exponent. In the general case, for example, for etc., this cannot be done, since the logarithm has an irrational value. Let us pay attention to one question related to this statement. In § 12 we gave the concept of the possibility of determining any real power of a given positive number. This was necessary for the introduction of logarithms, which, in general, can be irrational numbers.
Consider some properties of logarithms.
Property 1. If the number and base are equal, then the logarithm is equal to one, and, conversely, if the logarithm is equal to one, then the number and base are equal.
Proof. Let By the definition of the logarithm, we have and whence
Conversely, let Then by definition
Property 2. The logarithm of unity to any base is equal to zero.
Proof. By the definition of the logarithm (the zero power of any positive base is equal to one, see (10.1)). From here
Q.E.D.
The converse statement is also true: if , then N = 1. Indeed, we have .
Before stating the following property of logarithms, we agree to say that two numbers a and b lie on the same side of a third number c if they are both either greater than c or less than c. If one of these numbers is greater than c and the other is less than c, then we say that they lie on opposite sides of c.
Property 3. If the number and base lie on the same side of unity, then the logarithm is positive; if the number and base lie on opposite sides of unity, then the logarithm is negative.
The proof of property 3 is based on the fact that the degree of a is greater than one if the base is greater than one and the exponent is positive, or the base is less than one and the exponent is negative. The degree is less than one if the base is greater than one and the exponent is negative, or the base is less than one and the exponent is positive.
There are four cases to be considered:
We confine ourselves to the analysis of the first of them, the reader will consider the rest on his own.
Let then the exponent in equality be neither negative nor equal to zero, therefore, it is positive, i.e., which was required to be proved.
Example 3. Find out which of the following logarithms are positive and which are negative:
Solution, a) since the number 15 and the base 12 are located on the same side of the unit;
b) , since 1000 and 2 are located on the same side of the unit; at the same time, it is not essential that the base is greater than the logarithmic number;
c), since 3.1 and 0.8 lie on opposite sides of unity;
G) ; why?
e) ; why?
The following properties 4-6 are often called the rules of logarithm: they allow, knowing the logarithms of some numbers, to find the logarithms of their product, quotient, degree of each of them.
Property 4 (the rule for the logarithm of the product). The logarithm of the product of several positive numbers in a given base is equal to the sum of the logarithms of these numbers in the same base.
Proof. Let positive numbers be given.
For the logarithm of their product, we write the equality (26.1) defining the logarithm:
From here we find
Comparing the exponents of the first and last expressions, we obtain the required equality:
Note that the condition is essential; the logarithm of the product of two negative numbers makes sense, but in this case we get
In general, if the product of several factors is positive, then its logarithm is equal to the sum of the logarithms of the modules of these factors.
Property 5 (quotient logarithm rule). The logarithm of a quotient of positive numbers is equal to the difference between the logarithms of the dividend and the divisor, taken in the same base. Proof. Consistently find
Q.E.D.
Property 6 (rule of the logarithm of the degree). The logarithm of the power of any positive number is equal to the logarithm of that number times the exponent.
Proof. We write again the main identity (26.1) for the number :
Q.E.D.
Consequence. The logarithm of the root of a positive number is equal to the logarithm of the root number divided by the exponent of the root:
We can prove the validity of this corollary by presenting how and using property 6.
Example 4. Logarithm to base a:
a) (it is assumed that all values b, c, d, e are positive);
b) (it is assumed that ).
Solution, a) It is convenient to pass in this expression to fractional powers:
Based on equalities (26.5)-(26.7) we can now write:
We notice that simpler operations are performed on the logarithms of numbers than on the numbers themselves: when multiplying numbers, their logarithms are added, when divided, they are subtracted, etc.
That is why logarithms have been used in computational practice (see Sec. 29).
The action inverse to the logarithm is called potentiation, namely: potentiation is the action by which this number itself is found by the given logarithm of a number. In essence, potentiation is not any special action: it comes down to raising the base to a power (equal to the logarithm of the number). The term "potentiation" can be considered synonymous with the term "exponentiation".
When potentiating, it is necessary to use the rules that are inverse to the rules of logarithm: replace the sum of logarithms with the logarithm of the product, the difference of logarithms with the logarithm of the quotient, etc. In particular, if there is any factor in front of the sign of the logarithm, then during potentiation it must be transferred to the indicator degrees under the sign of the logarithm.
Example 5. Find N if it is known that
Decision. In connection with the potentiation rule just stated, the factors 2/3 and 1/3, which are in front of the signs of logarithms on the right side of this equality, will be transferred to the exponents under the signs of these logarithms; we get
Now we replace the difference of logarithms with the logarithm of the quotient:
to obtain the last fraction in this chain of equalities, we freed the previous fraction from irrationality in the denominator (section 25).
Property 7. If the base is greater than one, then the larger number has a larger logarithm (and the smaller one has a smaller one), if the base is less than one, then the larger number has a smaller logarithm (and the smaller one has a larger one).
This property is also formulated as a rule for the logarithm of inequalities, both parts of which are positive:
When taking the logarithm of inequalities to a base greater than one, the inequality sign is preserved, and when taking a logarithm to a base less than one, the sign of the inequality is reversed (see also item 80).
The proof is based on properties 5 and 3. Consider the case when If , then and, taking the logarithm, we obtain
(a and N/M lie on the same side of unity). From here
Case a follows, the reader will figure it out for himself.
Logarithm of b (b > 0) to base a (a > 0, a ≠ 1) is the exponent to which you need to raise the number a to get b.
The base 10 logarithm of b can be written as log(b), and the logarithm to the base e (natural logarithm) - ln(b).
Often used when solving problems with logarithms:
Properties of logarithms
There are four main properties of logarithms.
Let a > 0, a ≠ 1, x > 0 and y > 0.
Property 1. Logarithm of the product
Logarithm of the product is equal to the sum of logarithms:
log a (x ⋅ y) = log a x + log a y
Property 2. Logarithm of the quotient
Logarithm of the quotient is equal to the difference of logarithms:
log a (x / y) = log a x – log a y
Property 3. Logarithm of the degree
Degree logarithm is equal to the product of the degree and the logarithm:
If the base of the logarithm is in the exponent, then another formula applies:
Property 4. Logarithm of the root
This property can be obtained from the property of the logarithm of the degree, since the root of the nth degree is equal to the power of 1/n:
The formula for going from a logarithm in one base to a logarithm in another base
This formula is also often used when solving various tasks for logarithms:
Special case:
Comparison of logarithms (inequalities)
Suppose we have 2 functions f(x) and g(x) under logarithms with the same bases and there is an inequality sign between them:
To compare them, you first need to look at the base of the logarithms a:
- If a > 0, then f(x) > g(x) > 0
- If 0< a < 1, то 0 < f(x) < g(x)
How to solve problems with logarithms: examples
Tasks with logarithms included in the USE in mathematics for grade 11 in task 5 and task 7, you can find tasks with solutions on our website in the appropriate sections. Also, tasks with logarithms are found in the bank of tasks in mathematics. You can find all examples by searching the site.
What is a logarithm
Logarithms have always been considered a difficult topic in the school mathematics course. There are many different definitions of the logarithm, but for some reason most textbooks use the most complex and unfortunate of them.
We will define the logarithm simply and clearly. Let's create a table for this:
So, we have powers of two.
Logarithms - properties, formulas, how to solve
If you take the number from the bottom line, then you can easily find the power to which you have to raise a two to get this number. For example, to get 16, you need to raise two to the fourth power. And to get 64, you need to raise two to the sixth power. This can be seen from the table.
And now - in fact, the definition of the logarithm:
base a of the argument x is the power to which the number a must be raised to get the number x.
Notation: log a x \u003d b, where a is the base, x is the argument, b is actually what the logarithm is equal to.
For example, 2 3 = 8 ⇒ log 2 8 = 3 (the base 2 logarithm of 8 is three because 2 3 = 8). Might as well log 2 64 = 6, because 2 6 = 64.
The operation of finding the logarithm of a number to a given base is called. So let's add a new row to our table:
| 2 1 | 2 2 | 2 3 | 2 4 | 2 5 | 2 6 |
| 2 | 4 | 8 | 16 | 32 | 64 |
| log 2 2 = 1 | log 2 4 = 2 | log 2 8 = 3 | log 2 16 = 4 | log 2 32 = 5 | log 2 64 = 6 |
Unfortunately, not all logarithms are considered so easily. For example, try to find log 2 5. The number 5 is not in the table, but logic dictates that the logarithm will lie somewhere on the segment. Because 2 2< 5 < 2 3 , а чем больше степень двойки, тем больше получится число.
Such numbers are called irrational: the numbers after the decimal point can be written indefinitely, and they never repeat. If the logarithm turns out to be irrational, it is better to leave it like this: log 2 5, log 3 8, log 5 100.
It is important to understand that the logarithm is an expression with two variables (base and argument). At first, many people confuse where the base is and where the argument is. To avoid annoying misunderstandings, just take a look at the picture:
Before us is nothing more than the definition of the logarithm. Remember: the logarithm is the power, to which you need to raise the base to get the argument. It is the base that is raised to a power - in the picture it is highlighted in red. It turns out that the base is always at the bottom! I tell this wonderful rule to my students at the very first lesson - and there is no confusion.
How to count logarithms
We figured out the definition - it remains to learn how to count logarithms, i.e. get rid of the "log" sign. To begin with, we note that two important facts follow from the definition:
- The argument and base must always be greater than zero. This follows from the definition of the degree by a rational exponent, to which the definition of the logarithm is reduced.
- The base must be different from unity, since a unit to any power is still a unit. Because of this, the question “to what power must one be raised to get two” is meaningless. There is no such degree!
Such restrictions are called valid range(ODZ). It turns out that the ODZ of the logarithm looks like this: log a x = b ⇒ x > 0, a > 0, a ≠ 1.
Note that there are no restrictions on the number b (the value of the logarithm) is not imposed. For example, the logarithm may well be negative: log 2 0.5 = −1, because 0.5 = 2 −1 .
However, now we are considering only numerical expressions, where it is not required to know the ODZ of the logarithm. All restrictions have already been taken into account by the compilers of the problems. But when logarithmic equations and inequalities come into play, the DHS requirements will become mandatory. Indeed, in the basis and argument there can be very strong constructions that do not necessarily correspond to the above restrictions.
Now consider the general scheme for calculating logarithms. It consists of three steps:
- Express the base a and the argument x as a power with the smallest possible base greater than one. Along the way, it is better to get rid of decimal fractions;
- Solve the equation for the variable b: x = a b ;
- The resulting number b will be the answer.
That's all! If the logarithm turns out to be irrational, this will be seen already at the first step. The requirement that the base be greater than one is very relevant: this reduces the likelihood of error and greatly simplifies calculations. Similarly with decimal fractions: if you immediately convert them to ordinary ones, there will be many times less errors.
Let's see how this scheme works on specific examples:
Task. Calculate the logarithm: log 5 25
- Let's represent the base and the argument as a power of five: 5 = 5 1 ; 25 = 52;
- Received an answer: 2.
Let's make and solve the equation:
log 5 25 = b ⇒(5 1) b = 5 2 ⇒5 b = 5 2 ⇒ b = 2;
Task. Calculate the logarithm:
Task. Calculate the logarithm: log 4 64
- Let's represent the base and the argument as a power of two: 4 = 2 2 ; 64 = 26;
- Let's make and solve the equation:
log 4 64 = b ⇒(2 2) b = 2 6 ⇒2 2b = 2 6 ⇒2b = 6 ⇒ b = 3; - Received an answer: 3.
Task. Calculate the logarithm: log 16 1
- Let's represent the base and the argument as a power of two: 16 = 2 4 ; 1 = 20;
- Let's make and solve the equation:
log 16 1 = b ⇒(2 4) b = 2 0 ⇒2 4b = 2 0 ⇒4b = 0 ⇒ b = 0; - Received a response: 0.
Task. Calculate the logarithm: log 7 14
- Let's represent the base and the argument as a power of seven: 7 = 7 1 ; 14 is not represented as a power of seven, because 7 1< 14 < 7 2 ;
- It follows from the previous paragraph that the logarithm is not considered;
- The answer is no change: log 7 14.
A small note on the last example. How to make sure that a number is not an exact power of another number? Very simple - just decompose it into prime factors. If there are at least two distinct factors in the expansion, the number is not an exact power.
Task. Find out if the exact powers of the number are: 8; 48; 81; 35; fourteen.
8 \u003d 2 2 2 \u003d 2 3 - the exact degree, because there is only one multiplier;
48 = 6 8 = 3 2 2 2 2 = 3 2 4 is not an exact power because there are two factors: 3 and 2;
81 \u003d 9 9 \u003d 3 3 3 3 \u003d 3 4 - exact degree;
35 = 7 5 - again not an exact degree;
14 \u003d 7 2 - again not an exact degree;
Note also that the prime numbers themselves are always exact powers of themselves.
Decimal logarithm
Some logarithms are so common that they have a special name and designation.
of the x argument is the base 10 logarithm, i.e. the power to which 10 must be raised to obtain x. Designation: lgx.
For example, log 10 = 1; log 100 = 2; lg 1000 = 3 - etc.
From now on, when a phrase like “Find lg 0.01” appears in the textbook, know that this is not a typo. This is the decimal logarithm. However, if you are not used to such a designation, you can always rewrite it:
log x = log 10 x
Everything that is true for ordinary logarithms is also true for decimals.
natural logarithm
There is another logarithm that has its own notation. In a sense, it is even more important than decimal. This is the natural logarithm.
of the x argument is the logarithm to the base e, i.e. the power to which the number e must be raised to get the number x. Designation: lnx.
Many will ask: what is the number e? This is an irrational number, its exact value cannot be found and written down. Here are just the first numbers:
e = 2.718281828459…
We will not delve into what this number is and why it is needed. Just remember that e is the base of the natural logarithm:
ln x = log e x
Thus ln e = 1; log e 2 = 2; ln e 16 = 16 - etc. On the other hand, ln 2 is an irrational number. In general, the natural logarithm of any rational number is irrational. Except, of course, unity: ln 1 = 0.
For natural logarithms, all the rules that are true for ordinary logarithms are valid.
See also:
Logarithm. Properties of the logarithm (power of the logarithm).
How to represent a number as a logarithm?
We use the definition of a logarithm.
The logarithm is an indicator of the power to which the base must be raised to get the number under the sign of the logarithm.
Thus, to represent a certain number c as a logarithm to the base a, you need to put a power with the same base as the base of the logarithm under the sign of the logarithm, and write this number c into the exponent:
In the form of a logarithm, you can represent absolutely any number - positive, negative, integer, fractional, rational, irrational:
In order not to confuse a and c in stressful conditions of a test or exam, you can use the following rule to remember:
what is below goes down, what is above goes up.
For example, you want to represent the number 2 as a logarithm to base 3.
We have two numbers - 2 and 3. These numbers are the base and the exponent, which we will write under the sign of the logarithm. It remains to determine which of these numbers should be written down, in the base of the degree, and which - up, in the exponent.
The base 3 in the record of the logarithm is at the bottom, which means that when we represent the deuce as a logarithm to the base of 3, we will also write 3 down to the base.
2 is higher than 3. And in the notation of the degree, we write the two above the three, that is, in the exponent:
Logarithms. First level.
Logarithms
logarithm positive number b by reason a, where a > 0, a ≠ 1, is the exponent to which the number must be raised. a, To obtain b.
Definition of logarithm can be briefly written like this:
This equality is valid for b > 0, a > 0, a ≠ 1. He is usually called logarithmic identity.
The action of finding the logarithm of a number is called logarithm.
Properties of logarithms:
The logarithm of the product:
Logarithm of the quotient from division:
Replacing the base of the logarithm:
Degree logarithm:
root logarithm:
Logarithm with power base:
Decimal and natural logarithms.
Decimal logarithm numbers call the base 10 logarithm of that number and write lg b
natural logarithm numbers call the logarithm of this number to the base e, where e is an irrational number, approximately equal to 2.7. At the same time, they write ln b.
Other Notes on Algebra and Geometry
Basic properties of logarithms
Basic properties of logarithms
Logarithms, like any number, can be added, subtracted and converted in every possible way. But since logarithms are not quite ordinary numbers, there are rules here, which are called basic properties.
These rules must be known - no serious logarithmic problem can be solved without them. In addition, there are very few of them - everything can be learned in one day. So let's get started.
Addition and subtraction of logarithms
Consider two logarithms with the same base: log a x and log a y. Then they can be added and subtracted, and:
- log a x + log a y = log a (x y);
- log a x - log a y = log a (x: y).
So, the sum of the logarithms is equal to the logarithm of the product, and the difference is the logarithm of the quotient. Please note: the key point here is - same grounds. If the bases are different, these rules do not work!
These formulas will help calculate the logarithmic expression even when its individual parts are not considered (see the lesson "What is a logarithm"). Take a look at the examples and see:
log 6 4 + log 6 9.
Since the bases of logarithms are the same, we use the sum formula:
log 6 4 + log 6 9 = log 6 (4 9) = log 6 36 = 2.
Task. Find the value of the expression: log 2 48 − log 2 3.
The bases are the same, we use the difference formula:
log 2 48 - log 2 3 = log 2 (48: 3) = log 2 16 = 4.
Task. Find the value of the expression: log 3 135 − log 3 5.
Again, the bases are the same, so we have:
log 3 135 − log 3 5 = log 3 (135: 5) = log 3 27 = 3.
As you can see, the original expressions are made up of "bad" logarithms, which are not considered separately. But after transformations quite normal numbers turn out. Many tests are based on this fact. Yes, control - similar expressions in all seriousness (sometimes - with virtually no changes) are offered at the exam.
Removing the exponent from the logarithm
Now let's complicate the task a little. What if there is a degree in the base or argument of the logarithm? Then the exponent of this degree can be taken out of the sign of the logarithm according to the following rules:
It is easy to see that the last rule follows their first two. But it's better to remember it anyway - in some cases it will significantly reduce the amount of calculations.
Of course, all these rules make sense if the ODZ logarithm is observed: a > 0, a ≠ 1, x > 0. And one more thing: learn to apply all formulas not only from left to right, but also vice versa, i.e. you can enter the numbers before the sign of the logarithm into the logarithm itself.
How to solve logarithms
This is what is most often required.
Task. Find the value of the expression: log 7 49 6 .
Let's get rid of the degree in the argument according to the first formula:
log 7 49 6 = 6 log 7 49 = 6 2 = 12
Task. Find the value of the expression:
Note that the denominator is a logarithm whose base and argument are exact powers: 16 = 2 4 ; 49 = 72. We have:
I think the last example needs clarification. Where have logarithms gone? Until the very last moment, we work only with the denominator. They presented the base and the argument of the logarithm standing there in the form of degrees and took out the indicators - they got a “three-story” fraction.
Now let's look at the main fraction. The numerator and denominator have the same number: log 2 7. Since log 2 7 ≠ 0, we can reduce the fraction - 2/4 will remain in the denominator. According to the rules of arithmetic, the four can be transferred to the numerator, which was done. The result is the answer: 2.
Transition to a new foundation
Speaking about the rules for adding and subtracting logarithms, I specifically emphasized that they only work with the same bases. What if the bases are different? What if they are not exact powers of the same number?
Formulas for transition to a new base come to the rescue. We formulate them in the form of a theorem:
Let the logarithm log a x be given. Then for any number c such that c > 0 and c ≠ 1, the equality is true:
In particular, if we put c = x, we get:
It follows from the second formula that it is possible to interchange the base and the argument of the logarithm, but in this case the whole expression is “turned over”, i.e. the logarithm is in the denominator.
These formulas are rarely found in ordinary numerical expressions. It is possible to evaluate how convenient they are only when solving logarithmic equations and inequalities.
However, there are tasks that cannot be solved at all except by moving to a new foundation. Let's consider a couple of these:
Task. Find the value of the expression: log 5 16 log 2 25.
Note that the arguments of both logarithms are exact exponents. Let's take out the indicators: log 5 16 = log 5 2 4 = 4log 5 2; log 2 25 = log 2 5 2 = 2log 2 5;
Now let's flip the second logarithm:
Since the product does not change from permutation of factors, we calmly multiplied four and two, and then figured out the logarithms.
Task. Find the value of the expression: log 9 100 lg 3.
The base and argument of the first logarithm are exact powers. Let's write it down and get rid of the indicators:
Now let's get rid of the decimal logarithm by moving to a new base:
Basic logarithmic identity
Often in the process of solving it is required to represent a number as a logarithm to a given base.
In this case, the formulas will help us:
In the first case, the number n becomes the exponent in the argument. The number n can be absolutely anything, because it's just the value of the logarithm.
The second formula is actually a paraphrased definition. It's called like this:
Indeed, what will happen if the number b is raised to such a degree that the number b in this degree gives the number a? That's right: this is the same number a. Read this paragraph carefully again - many people “hang” on it.
Like the new base conversion formulas, the basic logarithmic identity is sometimes the only possible solution.
Task. Find the value of the expression:
Note that log 25 64 = log 5 8 - just took out the square from the base and the argument of the logarithm. Given the rules for multiplying powers with the same base, we get:
If someone is not in the know, this was a real task from the Unified State Examination 🙂
Logarithmic unit and logarithmic zero
In conclusion, I will give two identities that are difficult to call properties - rather, these are consequences from the definition of the logarithm. They are constantly found in problems and, surprisingly, create problems even for "advanced" students.
- log a a = 1 is. Remember once and for all: the logarithm to any base a from that base itself is equal to one.
- log a 1 = 0 is. The base a can be anything, but if the argument is one, the logarithm is zero! Because a 0 = 1 is a direct consequence of the definition.
That's all the properties. Be sure to practice putting them into practice! Download the cheat sheet at the beginning of the lesson, print it out and solve the problems.
