Properties of logarithmation. What is logarithm

Area of \u200b\u200bpermissible values \u200b\u200b(odb) logarithm

Now let's talk about restrictions (OTZ - area permissible values variables).

We remember that, for example, square root It is impossible to extract from negative numbers; Or if we have fraction, the denominator cannot be zero. There are similar restrictions from logarithms:

That is, the argument, and the base should be greater than zero, and the base cannot be equal.

Why is that?

Let's start with simple: let's say that. Then, for example, the number does not exist, since in which the degree we did not build, always turns out. Moreover, it does not exist for any. But at the same time can be equal to anything (for the same reason, it is to either degree). Therefore, the object does not represent any interest, and it was simply thrown out of mathematics.

We have a similar problem in the case: in any positive degree, it is, and in negative it can not be erected at all, as it will be divided into zero (let's remind you that).

When we face the problem of the construction of a fractional degree (which is presented in the form of a root :. For example, (that is), but there is no.

Therefore, the negative reasons are easier to throw out than to mess with them.

Well, since the base is only a positive basis, then in which the degree we either erected it, we always get the number strictly positive. So the argument must be positive. For example, it does not exist, since in no degree will not be a negative number (and even zero, therefore does not exist too).

In tasks with logarithms, the first thing you need to write down the OTZ. I will give an example:

We solve the equation.

Recall the definition: logarithm is a degree in which the foundation must be issued to get an argument. And under the condition, this degree is equal to :.

We get ordinary quadratic equation:. I solve it with the help of the Vieta theorem: the amount of the roots is equal, and the work. Easy to pick up, these are numbers and.

But if you immediately take and record both of these numbers in response, you can get 0 points for the task. Why? Let's think that it will be if we substitute these roots into the initial equation?

This is clearly incorrect, as the base cannot be negative, that is, the root is "third-party".

To avoid such unpleasant privities, it is necessary to write down OTZ even before the solution began to solve the equation:

Then, having received the roots and, immediately throw the root, and write the right answer.

Example 1. (Try to solve yourself) :

Find the root of the equation. If the roots are somewhat, in the answer, specify a smaller one.

Decision:

First, write ...

Now I remember what is logarithm: what extent you need to build a reason to get an argument? In the second. I.e:

It would seem that the smaller root is equal. But this is not like this: According to the OST root - third-party, that is, it is not the root at all of this equation. Thus, the equation has only one root :.

Answer: .

Basic logarithmic identity

Recall the definition of logarithm in general form:

Substitute to the second equality instead of logarithm:

This equality is called the main logarithmic identity. Although in essence it is equality - just differently recorded definition of logarithm:

This is the degree in which you need to build to get.

For example:

Share more following examples:

Example 2.

Find the value of the expression.

Decision:

Recall the rule from the section:, that is, if the degree is raised into the degree, the indicators are multiplied. Apply it:

Example 3.

Prove that.

Decision:

Properties of logarithm

Unfortunately, the tasks are not always so simple - it is often necessary to simplify the expression, lead it to the usual mind, and only then it will be possible to calculate the value. It is easiest to do, knowing properties of logarithm. So let's learn the basic properties of logarithms. Each of them I will prove, because any rule is easier to remember if you know where it is taken from.

All these properties must be remembered, without them most tasks with logarithms will not be solved.

And now about all the properties of logarithms in more detail.

Property 1:

Evidence:

Let, then.

We have:, ppm

Property 2: Logarithm

The amount of logarithms with the same bases is equal to the logarithm of the work: .

Evidence:

Let, then. Let, then.

Example:Find the value of the expression :.

Decision: .

The newly learned formula helps to simplify the amount of logarithms, not the difference, so that these logarithms do not immediately combine. But you can do on the contrary - to "smash" the first logarithm for two: but the promised simplification:
.
Why do you need it? Well, for example: what is equal?

Now it is obvious that.

Now simplify itself:

Tasks:

Answers:

Property 3: Logarithm Difference:

Evidence:

Everything is exactly the same as in paragraph 2:

Let, then.

Let, then. We have:

An example from the past item is now becoming even easier:

An example is more complicated :. Guess yourself how to solve?

It should be noted here that we have no formula about logarithm in the square. This is something akin to expression - it is not easy to simplify.

Therefore, we distrace from the formula about the logarithm, and think about what kind of formulas do we use in mathematics most often? Still from grade 7!

It - . You need to get used to the fact that they are everywhere! And in the demonstrative, and in trigonometric, and in irrational tasks they meet. Therefore, they must be remembered.

If you look at the first two terms, it becomes clear that this square difference:

Reply to verification:

Simplification itself.

Examples

Answers.

Property 4: Executive degree from the argument of the logarithm:

Evidence:And here, too, we use the definition of logarithm: let, then. We have:, ppm

You can understand this rule like this:

That is, the degree of argument is made forward logarithm as a coefficient.

Example:Find the value of the expression.

Decision: .

Share myself:

Examples:

Answers:

Property 5: Executive degree from the base of the logarithm:

Evidence:Let, then.

We have:, ppm
Remember: out basis The degree is made out inverse Number, in contrast to the previous case!

Property 6: Executive degree from the base and argument of the logarithm:

Or if the extent is the same :.

Property 7: Transition to a new base:

Evidence:Let, then.

We have:, ppm

Property 8: Replacing places of base and argument of logarithm:

Evidence:This is a special case of formula 7: if we substitute, we get :, bt.d.

Consider a few more examples.

Example 4.

Find the value of the expression.

We use the property of logarithms No. 2 - the sum of logarithms with the same base is equal to the logarithm of the work:

Example 5.

Find the value of the expression.

Decision:

We use the property of logarithms No. 3 and No. 4:

Example 6.

Find the value of the expression.

Decision:

We use property number 7 - we turn to the base 2:

Example 7.

Find the value of the expression.

Decision:

How do you need an article?

If you read these lines, then you read the entire article.

And it is cool!

Now tell us how to you an article?

Have you learned to solve logarithms? If not, then what's the problem?

Write to us in the comments below.

And, yes, good luck on the exams.

On the exam and OGE and in general in life

(from the Greek λόγος - "word", "attitude" and ἀριθμός - "number") of the number b. Based on a. (Log α. b.) called such a number c., I. b.= a C.that is, entries log α b.=c. and b \u003d A. C. Equivalent. Logarithm makes sense if a\u003e 0, a ≠ 1, b\u003e 0.

Speaking in other words logarithm numbers b. Based on butis formulated as an indicator of the degree in which the number should be issued a.to get a number b.(Logarithm exists only in positive numbers).

From this formulation it follows that calculation X \u003d log α b.equivalent to solving the equation A x \u003d b.

For example:

log 2 8 \u003d 3 because 8 \u003d 2 3.

We highlight that the specified logarithm formulation makes it possible to immediately determine the value of logarithmWhen the number under the sign of the logarithm is some degree of foundation. And in the truth, the logarithm formulation makes it possible to justify that if b \u003d A withthen logarithm numbers b. Based on a. Raven from. It is also clear that the theme of logarithmation is closely interconnected with the topic the degree of number.

The logarithm calculation is called logarithming. Logarithmation is a mathematical operation of taking logarithm. When logarithming, the works of the factors are transformed in the amount of members.

Potentiation - This is a mathematical operation inverse logarithming. In the potentiation, the specified base is erected into the degree of expression on which potentiation is performed. At the same time, the amounts of members are transformed into the work of the factors.

Frequently used real logarithms with bases 2 (binary), eilera number E ≈ 2.718 (natural logarithm) and 10 (decimal).

At this stage it is advisable to consider samples of logarithmlog 7 2. , lN. √ 5, LG0.0001.

And the records of LG (-3), Log -3 3.2, log -1 -4.3 do not make sense, because in the first of them, a negative number is placed under the logarithm sign, in the second - a negative number Based on, and in the third - and a negative number under the sign of the logarithm and one at the base.

The conditions for determining the logarithm.

It is worth considering the conditions a\u003e 0, a ≠ 1, b\u003e 0, which are given definition of logarithm. Consider why these restrictions are taken. Equality of the form x \u003d log α will help us b. , called the basic logarithmic identity, which directly follows from the above definition of the logarithm.

Take the condition a ≠ 1.. Since the unit is either equal to one, then the equality x \u003d log α b. may exist only when b \u003d 1.But at the same time log 1 1 will be any actual number. To eliminate this ambiguity and takes a ≠ 1..

We prove the need for condition a\u003e 0.. For a \u003d 0. The logarithm formulation can only exist when b \u003d 0.. And, accordingly, then lOG 0 0.it can be any different number from zero, since zero in any nonzero degree is zero. Eliminate this ambiguity gives a condition a ≠ 0. And for a.<0 We would have to reject the analysis of the rational and irrational values \u200b\u200bof the logarithm, since the degree with a rational and irrational indicator is determined only for non-negative grounds. It is for this reason that condition agreed a\u003e 0..

And last condition b\u003e 0. It follows from inequality a\u003e 0.because x \u003d log α b., and the value of the degree with a positive basis a. always positively.

Features of logarithms.

Logarithmia Characterized by distinctive featureswhich caused their widespread use for significant relief of painstaking calculations. When moving "to the world of logarithms", multiplication is transformed into a significantly easier addition, division - for subtraction, and the construction of the root is transformed into multiplication and division into an indicator of the degree.

The wording of logarithms and the table of their values \u200b\u200b(for trigonometric functions) was first published in 1614 by Scottish mathematician John Necess. Logarithmic tables, enlarged and detailed by other scientists, were widely used in the implementation of scientific and engineering computing, and remained relevant yet electronic calculators and computers.

So, before us deducts. If you take a number from the bottom line, you can easily find a degree in which the deuce will have to be taken to get this number. For example, to get 16, you need two to build a fourth degree. And to get 64, you need two to build in the sixth degree. This is seen from the table.

And now - actually, the definition of logarithm:

The logarithm on the base A from the X argument is the degree in which the number A is to be taken to get the number x.

Designation: Log A x \u003d b, where A is the basis, X is an argument, B - actually, what is equal to logarithm.

For example, 2 3 \u003d 8 ⇒ Log 2 8 \u003d 3 (the logarithm for the base 2 from the number 8 is three, since 2 3 \u003d 8). With the same success Log 2 64 \u003d 6, since 2 6 \u003d 64.

The operation of finding the logarithm of the number for a given base is called logarithming. So, supplement our table with a new string:

2 1	2 2	2 3	2 4	2 5	2 6
2	4	8	16	32	64
log 2 2 \u003d 1	log 2 4 \u003d 2	log 2 8 \u003d 3	log 2 16 \u003d 4	log 2 32 \u003d 5	log 2 64 \u003d 6

Unfortunately, not all logarithms are considered so easy. For example, try to find Log 2 5. Numbers 5 No in the table, but logic suggests that logarithm will lie somewhere on the segment. Because 2 2< 5 < 2 3 , а чем more degree Twos, the more the number will turn out.

Such numbers are called irrational: the numbers after the comma can be written to infinity, and they never repeat. If the logarithm is obtained irrational, it is better to leave it: log 2 5, Log 3 8, log 5 100.

It is important to understand that logarithm is an expression with two variables (base and argument). Many at first confuse where the basis is located, and where is the argument. To avoid annoying misunderstandingsJust take a look at the picture:

Before us is nothing more than the definition of logarithm. Remember: logarithm is a degreeIn which the foundation must be taken to get an argument. It is the foundation that is being built into a degree - in the picture it is highlighted in red. It turns out that the base is always downstairs! This wonderful rule I tell my students at the first lesson - and no confusion arises.

We dealt with the definition - it remains to learn to consider logarithms, i.e. Get rid of the sign "Log". To begin with, we note that two important facts follow from the definition:

1. The argument and the base should always be greater than zero. This follows from determining the degree of rational indicator to which the definition of logarithm is reduced.
2. The base should be different from the unit, since the unit to either degree still remains unity. Because of this, the question "How much the unit should be erected to get a deuce" deprived of meaning. There is no such degree!

Such restrictions are called the area of \u200b\u200bpermissible values (OTZ). It turns out that odd logarithm looks like this: log a x \u003d b ⇒ x\u003e 0, a\u003e 0, a ≠ 1.

Note that no restrictions on the number B (the value of logarithm) is not superimposed. For example, logarithm may well be negative: Log 2 0.5 \u003d -1, because 0.5 \u003d 2 -1.

However, now we are considering only numerical expressions, where to know the OTZ logarithm is not required. All restrictions are already taken into account by the compilers of tasks. But when logarithmic equations and inequalities go, the requirements of OTZ will become mandatory. Indeed, at the base and argument, very unreasonable structures can be standing, which necessarily comply with the above limitations.

Now consider general scheme Calculations of logarithms. It consists of three steps:

1. Submit the base a and argument x in the form of a degree with the minimum possible base, a large unit. Along the way, it is better to get rid of decimal fractions;
2. Solve relative to the variable B Equation: X \u003d A B;
3. The resulting number B will be the answer.

That's all! If the logarithm is irrational, it will be visible in the first step. The requirement that the base was more united is very important: it reduces the likelihood of error and greatly simplifies the calculations. Similar to S. decimal fractions: If you immediately transfer them to ordinary, errors will be at times less.

Let's see how this scheme works on specific examples:

A task. Calculate logarithm: log 5 25

1. Present the basis and argument as the degree of five: 5 \u003d 5 1; 25 \u003d 5 2;
2. Let us and solve the equation:
   log 5 25 \u003d B ⇒ (5 1) b \u003d 5 2 ⇒ 5 b \u003d 5 2 ⇒ b \u003d 2;
3. Received the answer: 2.

A task. Calculate logarithm:

A task. Calculate Logarithm: LOG 4 64

1. Imagine the basis and argument as a degree of twos: 4 \u003d 2 2; 64 \u003d 2 6;
2. Let us and solve the equation:
   log 4 64 \u003d b ⇒ (2 2) b \u003d 2 6 ⇒ 2 2b \u003d 2 6 ⇒ 2b \u003d 6 ⇒ B \u003d 3;
3. Received the answer: 3.

A task. Calculate logarithm: log 16 1

1. Imagine the basis and argument as a degree of two: 16 \u003d 2 4; 1 \u003d 2 0;
2. Let us and solve the equation:
   log 16 1 \u003d b ⇒ (2 4) b \u003d 2 0 ⇒ 2 4b \u003d 2 0 ⇒ 4b \u003d 0 ⇒ b \u003d 0;
3. Received the answer: 0.

A task. Calculate Logarithm: Log 7 14

1. Present the basis and argument as a degree of seven: 7 \u003d 7 1; 14 In the form of the degree of seven, it does not seem, since 7 1< 14 < 7 2 ;
2. From the previous point it follows that logarithm is not considered;
3. The answer is no change: log 7 14.

Little remark to the last example. How to make sure that the number is not the exact degree of another number? Very simple - enough to decompose it on simple factors. And if such multipliers cannot be assembled to the extent with the same indicators, the initial number is not an accurate degree.

A task. Find out whether the exact degrees of the number: 8; 48; 81; 35; fourteen.

8 \u003d 2 · 2 · 2 \u003d 2 3 - accurate degree, because The multiplier is only one;
48 \u003d 6 · 8 \u003d 3 · 2 · 2 · 2 · 2 \u003d 3 · 2 4 - It is not an exact degree, since there are two factors: 3 and 2;
81 \u003d 9 · 9 \u003d 3 · 3 · 3 · 3 \u003d 3 4 - accurate degree;
35 \u003d 7 · 5 - again is not an accurate degree;
14 \u003d 7 · 2 - Again, not exact degree;

Note also that simple numbers Always are the exact degrees of themselves.

Decimal logarithm

Some logarithms are encountered as often that they have a special name and designation.

The decimal logarithm from the X argument is a logarithm based on 10, i.e. The degree in which the number 10 should be erected to get the number x. Designation: LG X.

For example, LG 10 \u003d 1; lg 100 \u003d 2; LG 1000 \u003d 3 - etc.

From now on, when the textbook encounters the phrase like "Find LG 0.01", know: it is not a typo. This is a decimal logarithm. However, if you are unusual for such a designation, it can always be rewritten:
lG X \u003d log 10 x

All that is true for ordinary logarithms is true for decimal.

Natural logarithm

There is another logarithm that has its own designation. In a sense, it is even more important than decimal. We are talking about natural logarithm.

Natural logarithm from the argument X is a logarithm based on E, i.e. The degree in which the number e should be erected to get the number x. Designation: LN X.

Many will ask: what else in the number e? it irrational numberIt is impossible to find it, it is impossible to find it. I will give only its first figures:
e \u003d 2,718281828459 ...

We will not deepen that this is the number and why you need. Just remember that E is the basis of the natural logarithm:
ln x \u003d log e x

Thus, Ln E \u003d 1; ln e 2 \u003d 2; LN E 16 \u003d 16 - etc. On the other hand, LN 2 is an irrational number. In general, the natural logarithm of any rational number is irrational. In addition, of course, units: ln 1 \u003d 0.

For natural logarithms, all the rules that are true for ordinary logarithms are valid.

The logarithm of the positive number B for the base A (A\u003e 0, A is not equal to 1) they call such a number with that A C \u003d B: Log A B \u003d C ⇔ A C \u003d B (A\u003e 0, A ≠ 1, B\u003e \u200b\u200b0) & NBSP & NBSP & NBSP & NBSP & NBSP & NBSP

Please note: the logarithm from an inadequate number is not defined. In addition, at the base of the logarithm should be a positive number, not equal to 1. For example, if we are erected in a square, we obtain the number 4, but this does not mean that the logarithm on the base is -2 from 4 is 2.

Basic logarithmic identity

a log a b \u003d b (a\u003e 0, a ≠ 1) (2)

It is important that the areas of determining the right and left parts of this formula are different. Left part It is determined only at b\u003e 0, a\u003e 0 and a ≠ 1. The right side is defined at any b, and it does not depend on A at all. Thus, the use of the main logarithmic "identity" in solving equations and inequalities can lead to a change in the OTZ.

Two obvious consequences of the definition of logarithm

Log A A \u003d 1 (A\u003e 0, A ≠ 1) (3)
Log A 1 \u003d 0 (A\u003e 0, A ≠ 1) (4)

Indeed, when the number A is erected in the first degree, we will get the same number, and when it is erected into a zero degree.

Logarithm works and logarithm private

Log A (B C) \u003d Log A B + Log A C (A\u003e 0, A ≠ 1, B\u003e \u200b\u200b0, C\u003e 0) (5)

Log a b c \u003d log a b - log a c (a\u003e 0, a ≠ 1, b\u003e 0, c\u003e 0) (6)

I would like to warn schoolchildren from thoughtless application of these formulas in solving logarithmic equations and inequalities. When using them, "from left to right" there is a narrowing of OTZ, and in the transition from the amount or difference of logarithms to the logarithm of the work or private - the expansion of OTZ.

Indeed, the expression Log A (F (X) G (X)) is defined in two cases: when both functions are strictly positive or when f (x) and g (x) are less than zero.

Converting this expression in the amount of log a f (x) + Log A G (x), we are forced to limitate only by the case when f (x)\u003e 0 and g (x)\u003e 0. There is a narrowing area of \u200b\u200bpermissible values, and this is categorically unacceptable, since it can lead to loss of decisions. A similar problem exists for formula (6).

The degree can be made for the logarithm sign

Log A B P \u003d P Log A B (A\u003e 0, A ≠ 1, b\u003e 0) (7)

And again I would like to call for accuracy. Consider the following example:

Log A (F (X) 2 \u003d 2 Log A F (X)

The left part of equality is determined, obviously, with all values \u200b\u200bof F (x), except for zero. Right part - only at F (X)\u003e 0! After making a degree from the logarithm, we suvain the OTZ. The reverse procedure leads to expanding the area of \u200b\u200bpermissible values. All these comments refer not only to degree 2, but also to any even degree.

Formula of the transition to a new base

Log a B \u003d log c b log c a (a\u003e 0, a ≠ 1, b\u003e 0, c\u003e 0, c ≠ 1) (8)

The rare case when OTZ does not change when converting. If you wisely chose the base with (positive and not equal to 1), the transition formula to a new base is absolutely safe.

If as a new base with choose the number B, we get an important special case of formula (8):

Log A B \u003d 1 Log B A (A\u003e 0, A ≠ 1, B\u003e \u200b\u200b0, B ≠ 1) (9)

Some simple examples with logarithms

Example 1. Calculate: LG2 + LG50.
Decision. LG2 + LG50 \u003d LG100 \u003d 2. We used the formula sum of logarithms (5) and the determination of the decimal logarithm.

Example 2. Calculate: LG125 / LG5.
Decision. LG125 / LG5 \u003d log 5 125 \u003d 3. We used the transition to a new base (8).

Table formulas related to logarithms

A log a b \u003d b (a\u003e 0, a ≠ 1)

Log A A \u003d 1 (A\u003e 0, A ≠ 1)

Log A 1 \u003d 0 (A\u003e 0, A ≠ 1)

log a (b c) \u003d log a b + log a c (a\u003e 0, a ≠ 1, b\u003e 0, c\u003e 0)

Log A B C \u003d Log A B - Log A C (A\u003e 0, A ≠ 1, B\u003e \u200b\u200b0, C\u003e 0)

Log A B P \u003d P Log A B (A\u003e 0, A ≠ 1, b\u003e 0)

Log a B \u003d log c b log c a (a\u003e 0, a ≠ 1, b\u003e 0, c\u003e 0, c ≠ 1)

Log A B \u003d 1 Log B A (A\u003e 0, A ≠ 1, b\u003e 0, b ≠ 1)

The focus of this article - logarithm. Here we will give the definition of logarithm, show the adopted designation, we give examples of logarithms, and let's say about natural and decimal logarithms. After that, consider the main logarithmic identity.

Navigating page.

Definition of logarithm

The concept of logarithm occurs when solving the problem in a certain sense Reverse when it is necessary to find an indicator of a certain value of the degree and a well-known basis.

But enough prefaces, it's time to answer the question "What is logarithm? Let's give the appropriate definition.

Definition.

Logarithm number B based, where a\u003e 0, a ≠ 1 and b\u003e 0 is an indicator of the degree in which the number A is to be erected in order to obtain b.

At this stage, we note that the pronounced word "logarithm" should immediately call the resulting question: "What is the number" and "on what basis". In other words, just a logarithm as it were, and there is only a logarithm of numbers on some reason.

Immediately introduce designation of logarithm: The logarithm of the number B based on A is taken to be denoted as Log A B. The logarithm of the number B based on E and the logarithm based on the base 10 has its own special designations of LNB and LGB, respectively, that is, not the log e b, but lnb, and not log 10 b, and LGB.

Now you can give :.
And records It makes no sense, since in the first of them, under the sign of the logarithm there is a negative number, in the second - a negative number at the base, and in the third - and a negative number under the sign of the logarithm and one at the base.

Now let's say O. logarovmov reading rules. Log a B recording is read as "Logarithm B based on A". For example, Log 2 3 is a logarithm of three on the base 2, and is the logarithm of two integer two thirds on the base square root out of five. Logarithm based on E called natural logarithmAnd LNB recording is read as "natural logarithm B". For example, LN7 is a natural logarithm of seven, and we will read as a natural logarithm pi. Logarithm based on the base 10 also has a special name - decimal logarithmAnd the LGB record is read as the "decimal logarithm B". For example, LG1 is a decimal logarithm unit, and LG2,75 is a decimal logarithm of two whole seventy-five hundredths.

It is worth it separately on the terms a\u003e 0, a ≠ 1 and b\u003e 0, under which the definition of logarithm is given. Let us explain where these restrictions come from. Make it will help us equality of the species called, which directly follows from the above definition of the logarithm.

Let's start with a ≠ 1. Since the unit is to any degree equal to one, the equality can be valid only at B \u003d 1, but the Log 1 1 can be any valid number. To avoid this multi-rival and is accepted A ≠ 1.

Let's justify the expediency of condition a\u003e 0. At a \u003d 0, by definition of the logarithm, we would have equality that is possible only at B \u003d 0. But then log 0 0 can be any different number different from zero, as zero in any non-zero degree is zero. Avoid this multi-rival allows condition a ≠ 0. And with 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\u003e 0 follows from inequality a\u003e 0, since, and the value of a degree with a positive base A is always positive.

In conclusion of this item, let's say that the voiced definition of the logarithm allows you to immediately specify the value of the logarithm when the number under the logarithm sign is some degree of foundation. Indeed, the definition of a logarithm allows you to assert that if B \u003d a p, then the logarithm of the number B for the base A is equal to p. That is, the equality Log A A p \u003d p is valid. For example, we know that 2 3 \u003d 8, then log 2 8 \u003d 3. We will talk about this in more detail in the article.