(from Greek λόγος - “word”, “relation” and ἀριθμός - “number”) numbers b based on a(log α b) is called such a number c, And b= a c, that is, records 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 based on A formulated as an 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 α b, is equivalent to solving the equation a x =b.
For example:
log 2 8 = 3 because 8 = 2 3 .
Let us emphasize that the indicated formulation of the logarithm makes it possible to immediately determine logarithm value, when the number under the logarithm sign acts as 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 based on a equals With. It is also clear that the topic of logarithms is closely related to the topic powers of a number.
Calculating the logarithm is called logarithm. Logarithm is the mathematical operation of taking a logarithm. When taking logarithms, products of factors are transformed into sums of terms.
Potentiation is the inverse mathematical operation of logarithm. During potentiation, a given base is raised to the degree of expression over which potentiation is performed. In this case, the sums of terms are transformed into a product of factors.
Real logarithms with bases 2 (binary) are quite often used, e Euler number e ≈ 2.718 ( natural logarithm) and 10 (decimal).
At this stage it is advisable to consider logarithm samples 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 there is a negative number in the base, and in the third there is a negative number under the logarithm sign and unit at the base.
Conditions for determining the logarithm.
It is worth considering separately the conditions a > 0, a ≠ 1, b > 0.under which we get definition of logarithm. Let's look at why these restrictions were taken. An equality of the form x = log α will help us with this b, called the basic logarithmic identity, which directly follows from the definition of logarithm given above.
Let's take the condition a≠1. Since one to any power is equal to one, 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 exist only when b=0. And accordingly then log 0 0 can be any non-zero real number, since zero to any non-zero power is zero. This ambiguity can be eliminated by the condition a≠0. And when a<0 we would have to reject the analysis of rational and irrational values of the logarithm, since a degree with a rational and irrational exponent is defined only for non-negative bases. It is for this reason that the condition is stipulated a>0.
And the last condition b>0 follows from inequality a>0, since 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 significantly facilitate painstaking calculations. When moving “into the world of logarithms,” multiplication is transformed into a much easier addition, division is transformed into subtraction, and exponentiation and root extraction are transformed, respectively, into multiplication and division by the exponent.
Formulation of logarithms and 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 the use of electronic calculators and computers.
Range of acceptable values (APV) of the logarithm
Now let's talk about restrictions (ODZ - the range of permissible values of variables).
We remember that, for example, square root cannot be extracted from negative numbers; or if we have a fraction, then the denominator cannot be equal to zero. Logarithms have similar limitations:
That is, both the argument and the base must be greater than zero, but the base cannot yet be equal.
Why is this so?
Let's start with a simple thing: let's say that. Then, for example, the number does not exist, since no matter what power we raise to, it always turns out. Moreover, it does not exist for anyone. But at the same time it can be equal to anything (for the same reason - it is equal to any degree). Therefore, the object is of no interest, and it was simply thrown out of mathematics.
We have a similar problem in the case: in any positive degree- this, but it cannot be raised to negative at all, since it will result in division by zero (let me remind you that).
When we are faced with the problem of raising to a fractional power (which is represented as a root: . For example, (that is), but it does not exist.
Therefore, it is easier to throw away negative reasons than to mess with them.
Well, since our base a can only be positive, then no matter what power we raise it to, we will always get a strictly positive number. So the argument must be positive. For example, does not exist, since in no way will there be negative number(and even zero, therefore also does not exist).
In problems with logarithms, the first thing you need to do is write down the ODZ. Let me give you an example:
Let's solve the equation.
Let's remember the definition: a logarithm is the power to which the base must be raised to obtain an argument. And according to the condition, this degree is equal to: .
We get the usual quadratic equation: . Let's solve it using Vieta's theorem: the sum of the roots is equal, and the product. Easy to pick up, these are numbers and.
But if you immediately take and write both of these numbers in the answer, you can get 0 points for the problem. Why? Let's think about what happens if we substitute these roots into the initial equation?
This is clearly incorrect, since the base cannot be negative, that is, the root is “third party”.
To avoid such unpleasant pitfalls, you need to write down the ODZ even before starting to solve the equation:
Then, having received the roots and, we immediately discard the root and write the correct answer.
Example 1(try to solve it yourself) :
Find the root of the equation. If there are several roots, indicate the smallest of them in your answer.
Solution:
First of all, let’s write the ODZ:
Now let's remember what a logarithm is: to what power do you need to raise the base to get the argument? To the second. That is:
It would seem that the smaller root is equal. But this is not so: according to the ODZ, the root is extraneous, that is, it is not the root of this equation at all. Thus, the equation has only one root: .
Answer: .
Basic logarithmic identity
Let us recall the definition of logarithm in general form:
Let's substitute the logarithm into the second equality:
This equality is called basic logarithmic identity. Although in essence this is equality - just written differently definition of logarithm:
This is the power to which you must raise to get.
For example:
Solve the following examples:
Example 2.
Find the meaning of the expression.
Solution:
Let us remember the rule from the section:, that is, when raising a power to a power, the exponents are multiplied. Let's apply it:
Example 3.
Prove that.
Solution:
Properties of logarithms
Unfortunately, the tasks are not always so simple - often you first need to simplify the expression, bring it to its usual form, and only then will it be possible to calculate the value. This is easiest to do if you know properties of logarithms. So let's learn the basic properties of logarithms. I will prove each of them, because any rule is easier to remember if you know where it comes from.
All these properties must be remembered; without them, most problems with logarithms cannot be solved.
And now about all the properties of logarithms in more detail.
Property 1:
Proof:
Let it be then.
We have: , etc.
Property 2: Sum of logarithms
The sum of logarithms with the same bases is equal to the logarithm of the product: .
Proof:
Let it be then. Let it be then.
Example: Find the meaning of the expression: .
Solution: .
The formula you just learned helps to simplify the sum of logarithms, not the difference, so these logarithms cannot be combined right away. But you can do the opposite - “split” the first logarithm into two: And here is the promised simplification:
.
Why is this necessary? Well, for example: what does it equal?
Now it's obvious that.
Now simplify it yourself:
Tasks:
Answers:
Property 3: Difference of logarithms:
Proof:
Everything is exactly the same as in point 2:
Let it be then.
Let it be then. We have:
The example from the previous paragraph now becomes even simpler:
A more complicated example: . Can you figure out how to solve it yourself?
Here it should be noted that we do not have a single formula about logarithms squared. This is something akin to an expression - it cannot be simplified right away.
Therefore, let’s take a break from formulas about logarithms and think about what kind of formulas we use in mathematics most often? Since 7th grade!
This - . You need to get used to the fact that they are everywhere! And in exponential, and in trigonometric, and in irrational problems they meet. Therefore, they must be remembered.
If you look closely at the first two terms, it becomes clear that this difference of squares:
Answer to check:
Simplify it yourself.
Examples
Answers.
Property 4: Taking the exponent out of the logarithm argument:
Proof: And here we also use the definition of logarithm: let, then. We have: , etc.
This rule can be understood this way:
That is, the degree of the argument is moved ahead of the logarithm as a coefficient.
Example: Find the meaning of the expression.
Solution: .
Decide for yourself:
Examples:
Answers:
Property 5: Taking the exponent from the base of the logarithm:
Proof: Let it be then.
We have: , etc.
Remember: from grounds the degree is expressed as the opposite number, unlike the previous case!
Property 6: Removing the exponent from the base and argument of the logarithm:
Or if the degrees are the same: .
Property 7: Transition to a new base:
Proof: Let it be then.
We have: , etc.
Property 8: Swap the base and argument of the logarithm:
Proof: This special case formulas 7: if we substitute, we get: , etc.
Let's look at a few more examples.
Example 4.
Find the meaning of the expression.
We use property of logarithms No. 2 - the sum of logarithms with the same base is equal to the logarithm of the product:
Example 5.
Find the meaning of the expression.
Solution:
We use the property of logarithms No. 3 and No. 4:
Example 6.
Find the meaning of the expression.
Solution:
Let's use property No. 7 - move on to base 2:
Example 7.
Find the meaning of the expression.
Solution:
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On the Unified State Examination and the Unified State Examination and in life in general
Logarithms, like any numbers, can be added, subtracted and transformed in every way. But since logarithms are not exactly ordinary numbers, there are rules here, which are called main properties.
You definitely need to know these rules - not a single serious logarithmic problem can be solved without them. In addition, there are very few of them - you can learn everything in one day. So let's get started.
Adding and subtracting logarithms
Consider two logarithms with the same bases: 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 logarithms is equal to the logarithm of the product, and the difference is equal to the logarithm of the quotient. Please note: the key point here is identical grounds. If the reasons are different, these rules do not work!
These formulas will help you calculate logarithmic expression even when its individual parts are not considered (see lesson “ What is a logarithm"). Take a look at the examples and see:
Log 6 4 + log 6 9.
Since logarithms have the same bases, 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 calculated separately. But after the transformations, completely normal numbers are obtained. Many are built on this fact tests. Yes, test-like expressions are offered in all seriousness (sometimes with virtually no changes) on the Unified State Examination.
Extracting the exponent from the logarithm
Now let's complicate the task a little. What if the base or argument of a logarithm is a power? 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 the 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 of the 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 logarithm sign into the logarithm itself. 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 using the first formula:
log 7 49 6 = 6 log 7 49 = 6 2 = 12
Task. Find the meaning of the expression:
[Caption for the picture]
Note that the denominator contains a logarithm, the base and argument of which are exact powers: 16 = 2 4 ; 49 = 7 2. We have:
[Caption for the picture] I think to last example clarification required. Where have logarithms gone? Until the very last moment we work only with the denominator. We presented the base and argument of the logarithm standing there in the form of powers and took out the exponents - we got a “three-story” fraction.
Now let's look at the main fraction. The numerator and denominator contain 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 is what was done. The result was 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 reasons are different? What if they are not exact powers of the same number?
Formulas for transition to a new foundation come to the rescue. Let us formulate them in the form of a theorem:
Let the logarithm log be given a x. Then for any number c such that c> 0 and c≠ 1, the equality is true:
[Caption for the picture]
In particular, if we put c = x, we get:
[Caption for the picture]
From the second formula it follows that the base and argument of the logarithm can be swapped, but in this case the entire expression is “turned over”, i.e. the logarithm appears in the denominator.
These formulas are rarely found in conventional numerical expressions. It is possible to evaluate how convenient they are only by deciding logarithmic equations and inequalities.
However, there are problems that cannot be solved at all except by moving to a new foundation. Let's look at a couple of these:
Task. Find the value of the expression: log 5 16 log 2 25.
Note that the arguments of both logarithms contain exact powers. 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 “reverse” the second logarithm:
[Caption for the picture]Since the product does not change when rearranging factors, we calmly multiplied four and two, and then dealt with 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 this down and get rid of the indicators:
[Caption for the picture]Now let's get rid of decimal logarithm, moving to a new base:
[Caption for the picture]Basic logarithmic identity
Often in the solution process it is necessary to represent a number as a logarithm to a given base. In this case, the following formulas will help us:
In the first case, the number n becomes an indicator of the degree standing in the argument. Number n can be absolutely anything, because it’s just a logarithm value.
The second formula is actually a paraphrased definition. That’s what it’s called: the basic logarithmic identity.
In fact, what will happen if the number b raise to such a power that the number b to this power gives the number a? That's right: you get this same number a. Read this paragraph carefully again - many people get stuck on it.
Like formulas for moving to a new base, the basic logarithmic identity is sometimes the only possible solution.
Task. Find the meaning of the expression:
[Caption for the picture]
Note that log 25 64 = log 5 8 - we simply took the square from the base and argument of the logarithm. Taking into account the rules for multiplying powers with the same base, we get:
[Caption for the picture]If anyone doesn't know, this was a real task from the Unified State Exam :)
Logarithmic unit and logarithmic zero
In conclusion, I will give two identities that can hardly be called properties - rather, they are consequences of the definition of the logarithm. They constantly appear in problems and, surprisingly, create problems even for “advanced” students.
- log a a= 1 is a logarithmic unit. Remember once and for all: logarithm to any base a from this very base is equal to one.
- log a 1 = 0 is logarithmic zero. Base a can be anything, but if the argument contains one, the logarithm is equal to 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.
“Abbreviated multiplication formulas” - When multiplying two polynomials, each term of the first polynomial is multiplied by each term of the second polynomial and the products are added. Abbreviated multiplication formulas. When adding and subtracting polynomials, the rules for opening parentheses are used. Monomials are products of numbers, variables and their natural powers.
“Solving a system of equations” - Graphic method(algorithm). An equation is an equality containing one or more variables. Equation and its properties. Method of determinants (algorithm). System of equations and its solution. Solving the system using the comparison method. Linear equation with two variables. Solving the system using the addition method.
“Solving systems of inequalities” - Intervals. Mathematical dictation. Examples of solving systems are considered linear inequalities. Solving systems of inequalities. To solve a system of linear inequalities, it is enough to solve each of the inequalities included in it and find the intersection of the sets of their solutions. Write down inequalities whose solution sets are intervals.
“Exemplary inequalities” - Sign of inequality. Solve the inequality. The simplest solution exponential inequalities. Solving exponential inequalities. What should you consider when solving exponential inequalities? Solving simple exponential inequalities. An inequality containing an unknown exponent is called an exponential inequality.
“Number relationships” - What is proportion? What are the numbers m and n called in the proportion a: m = n: b? The quotient of two numbers is called the ratio of two numbers. Marketing Lan. In the correct proportion, the product of the extreme terms is equal to the product of the middle terms and vice versa. What is attitude? Proportions. The ratio can be expressed as a percentage.
"Discriminant of a quadratic equation" - Vieta's theorem. Quadratic equations. Discriminant. What equations are called incomplete quadratic equations? How many roots does an equation have if its discriminant is zero? Solving incomplete quadratic equations. How many roots does an equation have if its discriminant is a negative number?
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