Follows from its definition. And so the logarithm of the number b based on A is defined as the exponent to which a number must be raised a to get the number b(logarithm exists only for positive numbers).
From this formulation it follows that the calculation x=log a b, is equivalent to solving the equation a x =b. For example, log 2 8 = 3 because 8 = 2 3 . The formulation of the logarithm makes it possible to justify that if b=a c, then the logarithm of the number b based on a equals With. It is also clear that the topic of logarithms is closely related to the topic of powers of a number.
With logarithms, as with any numbers, you can do operations of addition, subtraction and transform in every possible way. But due to the fact that logarithms are not entirely ordinary numbers, their own special rules apply here, which are called main properties.
Adding and subtracting logarithms.
Let's take two logarithms with on the same grounds: log a x And log a y. Then it is possible to perform addition and subtraction operations:
log a x+ log a y= log a (x·y);
log a x - log a y = log a (x:y).
log a(x 1 . x 2 . x 3 ... x k) = log a x 1 + log a x 2 + log a x 3 + ... + log a x k.
From logarithm quotient theorem One more property of the logarithm can be obtained. It is common knowledge that log a 1= 0, therefore
log a 1 /b=log a 1 - log a b= -log a b.
This means there is an equality:
log a 1 / b = - log a b.
Logarithms of two reciprocal numbers for the same reason will differ from each other solely by sign. So:
Log 3 9= - log 3 1 / 9 ; log 5 1 / 125 = -log 5 125.
The focus of this article is logarithm. Here we will give the definition of logarithm, show accepted designation, we will give examples of logarithms, and talk about natural and decimal logarithms. After this we will consider the basic logarithmic identity.
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Definition of logarithm
The concept of a logarithm arises when solving a problem in in a certain sense inverse, when you need to find the exponent of known value degree and known basis.
But enough prefaces, it’s time to answer the question “what is a logarithm”? Let us give the corresponding 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 follow-up questions: “what number” and “on what basis.” In other words, there is simply no logarithm, but only the logarithm of a number to some base.
Let's enter right away logarithm notation: the logarithm of a number b to base a is usually denoted as log a b. The logarithm of a number b to base e and the logarithm to base 10 have their own special designations lnb and logb, respectively, that is, they write not log e b, but lnb, and not log 10 b, but lgb.
Now we can give: .
And the records 
do not make sense, since in the first of them there is a negative number under the logarithm sign, in the second there is a negative number in the base, and in the third there is a negative number under the logarithm sign and a unit in the base.
Now let's talk about rules for reading logarithms. Log a b is read as "the 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 point two thirds to base 2 Square root out of five. The logarithm to base e is called natural logarithm, and the notation lnb reads "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 base 10 logarithm also has a special name - decimal logarithm, and lgb is read as "decimal logarithm of b". For example, lg1 is the decimal logarithm of one, and lg2.75 is the decimal logarithm of two point seven 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. An equality of the form called , which directly follows from the definition of logarithm given above, will help us do this.
Let's start with a≠1. Since one to any power is equal to one, the equality can only be true when b=1, but log 1 1 can be any real number. To avoid this ambiguity, a≠1 is assumed.
Let us justify the expediency of the condition a>0. With a=0, by the definition of a logarithm, we would have equality, which is only possible with b=0. But then log 0 0 can be any non-zero real number, since zero to any non-zero power is zero. The condition a≠0 allows us to avoid this ambiguity. And when a<0 нам бы пришлось отказаться от рассмотрения рациональных и иррациональных значений логарифма, так как степень с рациональным и ирrational indicator defined only for non-negative bases. Therefore, the condition a>0 is accepted.
Finally, the condition b>0 follows from the inequality a>0, since , and the value of a power with a positive base a is always positive.
To conclude this point, let’s say that the stated definition of the logarithm allows you to immediately indicate the value of the logarithm when the number under the logarithm sign is a certain power of the base. Indeed, the definition of a logarithm allows us to state that if b=a p, then the logarithm of the number b to 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.
As you know, when multiplying expressions with powers, their exponents always add up (a b *a c = a b+c). This mathematical law was derived by Archimedes, and later, in the 8th century, the mathematician Virasen created a table of integer exponents. They were the ones who served for further opening logarithms. Examples of using this function can be found almost everywhere where you need to simplify cumbersome multiplication by simple addition. If you spend 10 minutes reading this article, we will explain to you what logarithms are and how to work with them. In simple and accessible language.
Definition in mathematics
A logarithm is an expression of the following form: log a b=c, that is, the logarithm of any non-negative number(that is, any positive) “b” by its base “a” is considered to be the power of “c” to which the base “a” must be raised in order to ultimately obtain the value “b”. Let's analyze the logarithm using examples, let's say there is an expression log 2 8. How to find the answer? It’s very simple, you need to find a power such that from 2 to the required power you get 8. After doing some calculations in your head, we get the number 3! And that’s true, because 2 to the power of 3 gives the answer as 8.
Types of logarithms
For many pupils and students, this topic seems complicated and incomprehensible, but in fact logarithms are not so scary, the main thing is to understand their general meaning and remember their properties and some rules. There are three individual species logarithmic expressions:
- Natural logarithm ln a, where the base is the Euler number (e = 2.7).
- Decimal a, where the base is 10.
- Logarithm of any number b to base a>1.
Each of them is decided in a standard way, which includes simplification, reduction and subsequent reduction to one logarithm using logarithmic theorems. To obtain the correct values of logarithms, you should remember their properties and the sequence of actions when solving them.
Rules and some restrictions
In mathematics, there are several rules-constraints that are accepted as an axiom, that is, they are not subject to discussion and are the truth. For example, it is impossible to divide numbers by zero, and it is also impossible to extract an even root from negative numbers. Logarithms also have their own rules, following which you can easily learn to work even with long and capacious logarithmic expressions:
- The base “a” must always be greater than zero, and not equal to 1, otherwise the expression will lose its meaning, because “1” and “0” to any degree are always equal to their values;
- if a > 0, then a b >0, it turns out that “c” must also be greater than zero.
How to solve logarithms?
For example, the task is given to find the answer to the equation 10 x = 100. This is very easy, you need to choose a power by raising the number ten to which we get 100. This, of course, is 10 2 = 100.
Now let's represent this expression in logarithmic form. We get log 10 100 = 2. When solving logarithms, all actions practically converge to find the power to which it is necessary to introduce the base of the logarithm in order to obtain given number.
To accurately determine the value unknown degree you need to learn how to work with the table of degrees. It looks like this:
As you can see, some exponents can be guessed intuitively if you have a technical mind and knowledge of the multiplication table. However for large values you will need a table of degrees. It can be used even by those who know nothing at all about complex mathematical topics. The left column contains numbers (base a), the top row of numbers is the value of the power c to which the number a is raised. At the intersection, the cells contain the number values that are the answer (a c =b). Let's take, for example, the very first cell with the number 10 and square it, we get the value 100, which is indicated at the intersection of our two cells. Everything is so simple and easy that even the most true humanist will understand!
Equations and inequalities
It turns out that under certain conditions the exponent is the logarithm. Therefore, any mathematical numerical expressions can be written as a logarithmic equation. For example, 3 4 =81 can be written as the base 3 logarithm of 81 equal to four (log 3 81 = 4). For negative powers the rules are the same: 2 -5 = 1/32 we write it as a logarithm, we get log 2 (1/32) = -5. One of the most fascinating sections of mathematics is the topic of “logarithms”. We will look at examples and solutions of equations below, immediately after studying their properties. Now let's look at what inequalities look like and how to distinguish them from equations.
Given an expression of the following form: log 2 (x-1) > 3 - it is logarithmic inequality, since the unknown value "x" is under the sign of the logarithm. And also in the expression two quantities are compared: the logarithm of the desired number to base two is greater than the number three.
The most important difference between logarithmic equations and inequalities is that equations with logarithms (for example, the logarithm 2 x = √9) imply one or more specific answers. numerical values, while when solving the inequalities are defined as the region acceptable values, and the breakpoints of this function. As a consequence, the answer is not a simple set of individual numbers, as in the answer to an equation, but rather continuous series or a set of numbers.
Basic theorems about logarithms
When solving primitive tasks of finding the values of the logarithm, its properties may not be known. However, when it comes to logarithmic equations or inequalities, first of all, it is necessary to clearly understand and apply in practice all the basic properties of logarithms. We will look at examples of equations later; let's first look at each property in more detail.
- The main identity looks like this: a logaB =B. It applies only when a is greater than 0, not equal to one, and B is greater than zero.
- The logarithm of the product can be represented in the following formula: log d (s 1 *s 2) = log d s 1 + log d s 2. In this case prerequisite is: d, s 1 and s 2 > 0; a≠1. You can give a proof for this logarithmic formula, with examples and solution. Let log a s 1 = f 1 and log a s 2 = f 2, then a f1 = s 1, a f2 = s 2. We obtain that s 1 * s 2 = a f1 *a f2 = a f1+f2 (properties of degrees ), and then by definition: log a (s 1 * s 2) = f 1 + f 2 = log a s1 + log a s 2, which is what needed to be proven.
- The logarithm of the quotient looks like this: log a (s 1/ s 2) = log a s 1 - log a s 2.
- The theorem in the form of a formula takes on next view: log a q b n = n/q log a b.
This formula is called the “property of the degree of logarithm.” It resembles the properties of ordinary degrees, and it is not surprising, because all mathematics is based on natural postulates. Let's look at the proof.
Let log a b = t, it turns out a t =b. If we raise both parts to the power m: a tn = b n ;
but since a tn = (a q) nt/q = b n, therefore log a q b n = (n*t)/t, then log a q b n = n/q log a b. The theorem has been proven.
Examples of problems and inequalities
The most common types of problems on logarithms are examples of equations and inequalities. They are found in almost all problem books, and are also included in mandatory part mathematics exams. For admission to university or passing entrance examinations in mathematics you need to know how to solve such problems correctly.
Unfortunately, there is no single plan or scheme for solving and determining unknown value There is no such thing as a logarithm, but you can apply it to every mathematical inequality or logarithmic equation. certain rules. First of all, you should find out whether the expression can be simplified or lead to general appearance. Simplify long ones logarithmic expressions possible if you use their properties correctly. Let's get to know them quickly.
When deciding logarithmic equations, we should determine what type of logarithm we have: an example expression may contain a natural logarithm or a decimal one.
Here are examples ln100, ln1026. Their solution boils down to the fact that they need to determine the power to which the base 10 will be equal to 100 and 1026, respectively. For solutions natural logarithms need to apply logarithmic identities or their properties. Let's look at the solution with examples logarithmic problems different types.
How to Use Logarithm Formulas: With Examples and Solutions
So, let's look at examples of using the basic theorems about logarithms.
- The property of the logarithm of a product can be used in tasks where it is necessary to expand great importance numbers b into simpler factors. For example, log 2 4 + log 2 128 = log 2 (4*128) = log 2 512. The answer is 9.
- log 4 8 = log 2 2 2 3 = 3/2 log 2 2 = 1.5 - as you can see, using the fourth property of the logarithm power, we managed to solve a seemingly complex and unsolvable expression. You just need to factor the base and then take the exponent values out of the sign of the logarithm.
Assignments from the Unified State Exam
Logarithms are often found in entrance exams, especially a lot of logarithmic problems in the Unified State Exam ( State exam for all school leavers). Typically, these tasks are present not only in part A (the easiest test part of the exam), but also in part C (the most complex and voluminous tasks). The exam requires accurate and perfect knowledge of the topic “Natural logarithms”.
Examples and solutions to problems are taken from official Unified State Exam options. Let's see how such tasks are solved.
Given log 2 (2x-1) = 4. Solution:
let's rewrite the expression, simplifying it a little log 2 (2x-1) = 2 2, by the definition of the logarithm we get that 2x-1 = 2 4, therefore 2x = 17; x = 8.5.
- It is best to reduce all logarithms to the same base so that the solution is not cumbersome and confusing.
- All expressions under the logarithm sign are indicated as positive, therefore, when the exponent of an expression that is under the logarithm sign and as its base is taken out as a multiplier, the expression remaining under the logarithm must be positive.
1.1. Determining the exponent for an integer exponent
X 1 = XX 2 = X * X
X 3 = X * X * X
…
X N = X * X * … * X — N times
1.2. Zero degree.
By definition, it is generally accepted that zero degree any number is equal to 1:1.3. Negative degree.
X -N = 1/X N1.4. Fractional power, root.
X 1/N = N root of X.For example: X 1/2 = √X.
1.5. Formula for adding powers.
X (N+M) = X N *X M1.6.Formula for subtracting powers.
X (N-M) = X N /X M1.7. Formula for multiplying powers.
X N*M = (X N) M1.8. Formula for raising a fraction to a power.
(X/Y) N = X N /Y N2. Number e.
The value of the number e is equal to the following limit:E = lim(1+1/N), as N → ∞.
With an accuracy of 17 digits, the number e is 2.71828182845904512.
3. Euler's equality.
This equality connects five numbers that play a special role in mathematics: 0, 1, e, pi, imaginary unit.E (i*pi) + 1 = 0
4. Exponential function exp(x)
exp(x) = e x5. Derivative of exponential function
The exponential function has remarkable property: The derivative of a function is equal to the exponential function itself:(exp(x))" = exp(x)
6. Logarithm.
6.1. Definition of the logarithm function
If x = b y, then the logarithm is the functionY = Log b(x).
The logarithm shows to what power a number must be raised - the base of the logarithm (b) to obtain a given number (X). The logarithm function is defined for X greater than zero.
For example: Log 10 (100) = 2.
6.2. Decimal logarithm
This is the logarithm to base 10:Y = Log 10 (x) .
Denoted by Log(x): Log(x) = Log 10 (x).
Usage example decimal logarithm- decibel.
6.3. Decibel
The item is highlighted on a separate page Decibel6.4. Binary logarithm
This is the base 2 logarithm:Y = Log 2 (x).
Denoted by Lg(x): Lg(x) = Log 2 (X)
6.5. Natural logarithm
This is the logarithm to base e:Y = Log e (x) .
Denoted by Ln(x): Ln(x) = Log e (X)
Natural logarithm - inverse function to exponential functions exp(X).
6.6. Characteristic points
Loga(1) = 0Log a (a) = 1
6.7. Product logarithm formula
Log a (x*y) = Log a (x)+Log a (y)6.8. Formula for logarithm of quotient
Log a (x/y) = Log a (x)-Log a (y)6.9. Logarithm of power formula
Log a (x y) = y*Log a (x)6.10. Formula for converting to a logarithm with a different base
Log b (x) = (Log a (x))/Log a (b)Example:
Log 2 (8) = Log 10 (8)/Log 10 (2) =
0.903089986991943552 / 0.301029995663981184 = 3
7. Formulas useful in life
Often there are problems of converting volume into area or length and inverse problem-- conversion of area to volume. For example, boards are sold in cubes (cubic meters), and we need to calculate how much wall area can be covered with boards contained in a certain volume, see calculation of boards, how many boards are in a cube. Or, if the dimensions of the wall are known, you need to calculate the number of bricks, see brick calculation.
It is permitted to use site materials provided that an active link to the source is installed.
In relation to
the task of finding any of the three numbers from the other two given ones can be set. If a and then N are given, they are found by exponentiation. If N and then a are given by taking the root of the degree x (or raising it to the power). Now consider the case when, given a and N, we need to find x.
Let the number N be positive: the number a be positive and not equal to one: .
Definition. The logarithm of the number N to the base a is the exponent to which a must be raised to obtain the number N; logarithm is denoted by
Thus, in equality (26.1) the exponent is found as the logarithm of N to base a. Posts
have same meaning. Equality (26.1) is sometimes called the main identity of the theory of logarithms; in reality it expresses the definition of the concept of logarithm. By this definition The base of the logarithm a is always positive and different from unity; the logarithmic number N is positive. Negative numbers and zero have no logarithms. It can be proven that any number with a given base has a well-defined logarithm. Therefore equality entails . Note that the essential condition here is otherwise the conclusion would not be justified, since the equality is true for any values of x and y.
Example 1. Find
Solution. To obtain a number, you must raise the base 2 to the power Therefore.
You can make notes when solving such examples in the following form:
Example 2. Find .
Solution. We have
In examples 1 and 2, we easily found the desired logarithm by representing the logarithm number as a power of the base with a rational exponent. IN general case, for example, for etc., this cannot be done, since the logarithm has irrational meaning. Let us pay attention to one issue related to this statement. In paragraph 12 we gave the concept of the possibility of determining any real degree of a given positive number. This was necessary for the introduction of logarithms, which, generally speaking, can be irrational numbers.
Let's look at some properties of logarithms.
Property 1. If the number and base are equal, then the logarithm 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 a logarithm we have and whence
Conversely, let Then by definition
Property 2. The logarithm of one to any base is equal to zero.
Proof. By definition of a 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 formulating the next property of logarithms, let us agree to say that two numbers a and b lie on the same side of the third number c if they are both 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 will say that they lie along different sides from the village
Property 3. If the number and base lie on the same side of one, then the logarithm is positive; If the number and base lie on opposite sides of one, then the logarithm is negative.
The proof of property 3 is based on the fact that the power 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. A power 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 consider:
We will limit ourselves to analyzing the first of them; the reader will consider the rest on his own.
Let then in equality the exponent can be neither negative nor equal to zero, therefore, it is positive, i.e., as required to be proved.
Example 3. Find out which of the logarithms below are positive and which are negative:
Solution, a) since the number 15 and the base 12 are located on the same side of one;
b) since 1000 and 2 are located on one side of the unit; in this case, it is not important that the base is greater than the logarithmic number;
c) since 3.1 and 0.8 lie on opposite sides of unity;
G) ; Why?
d) ; Why?
The following properties 4-6 are often called the rules of logarithmation: they allow, knowing the logarithms of some numbers, to find the logarithms of their product, quotient, and degree of each of them.
Property 4 (product logarithm rule). Logarithm of the product of several positive numbers by this basis equal to the sum logarithms of these numbers to the same base.
Proof. Let the given numbers be positive.
For the logarithm of their product, we write the equality (26.1) that defines the logarithm:
From here we will 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 absolute values of these factors.
Property 5 (rule for taking logarithms of quotients). The logarithm of a quotient of positive numbers is equal to the difference between the logarithms of the dividend and the divisor, taken to the same base. Proof. We consistently find
Q.E.D.
Property 6 (power logarithm rule). Logarithm of the power of some positive number equal to the logarithm this number multiplied by the exponent.
Proof. Let us write again the main identity (26.1) for the number:
Q.E.D.
Consequence. The logarithm of a root of a positive number is equal to the logarithm of the radical divided by the exponent of the root:
The validity of this corollary can be proven by imagining how and using property 6.
Example 4. Take 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 go to 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 dividing, they are subtracted, etc.
That is why logarithms are used in computing practice (see paragraph 29).
The inverse action of logarithm is called potentiation, namely: potentiation is the action by which the number itself is found from a given logarithm of a number. Essentially, potentiation is not any special action: it comes down to raising a base to a power ( equal to the logarithm numbers). The term "potentiation" can be considered synonymous with the term "exponentiation".
When potentiating, you must use the rules inverse to the rules of logarithmation: 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 a factor in front of the sign of the logarithm, then during potentiation it must be transferred to the exponent degrees under the sign of the logarithm.
Example 5. Find N if it is known that
Solution. In connection with the just stated rule of potentiation, we will transfer the factors 2/3 and 1/3 standing in front of the signs of logarithms on the right side of this equality into 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 (clause 25).
Property 7. If the base is greater than one, then larger number has a larger logarithm (and a smaller number has a smaller one), if the base is less than one, then a larger number has a smaller logarithm (and a smaller number has a larger one).
This property is also formulated as a rule for taking logarithms of inequalities, both sides of which are positive:
When taking logarithms of inequalities to the base, greater than one, the sign of inequality is preserved, and when taking a logarithm to a base less than one, the sign of inequality changes to the opposite (see also paragraph 80).
The proof is based on properties 5 and 3. Consider the case when If , then and, taking logarithms, we obtain
(a and N/M lie on the same side of unity). From here
Case a follows, the reader will figure it out on his own.