Decimal logarithm numbers how to solve. Tasks from the EGE

Logarithm number B (b\u003e 0) based on A (A\u003e 0, A ≠ 1) - An indicator of the degree in which the number A should be taken to get b.

Logarithm number B based on 10 can be written as lG (B), and logarithm based on E (natural logarithm) - ln (b).

Often used when solving tasks with logarithms:

Properties of logarithm

There are four basic properties of logarithm.

Let a\u003e 0, a ≠ 1, x\u003e 0 and y\u003e 0.

Property 1. Logarithm works

Logarithm works equal to the sum of logarithms:

log A (X ⋅ Y) \u003d Log A x + Log A Y

Property 2. Private logarithm

Logarithm private equal to the difference in logarithms:

log a (x / y) \u003d log a x - Log A Y

Property 3. Logarithm

Logarithm degree equal to the work extent on logarithm:

If the foundation of the logarithm is in the degree, the other formula acts:

Property 4. Logarithm root

This property can be obtained from the properties of the logarithm of degree, since the root of N-th degree is equal to 1 / N:

The formula for the transition from the logarithm in one base to the logarithm with a different base

This formula is also often used when solving various tasks for logarithmia:

Private case:

Comparison of logarithms (inequality)

Let we have 2 functions f (x) and g (x) under logarithms with the same bases and between them is the sign of inequality:

To compare them, you must first look at the base of logarithms A:

If a\u003e 0, then f (x)\u003e G (x)\u003e 0
If 0< a < 1, то 0 < f(x) < g(x)

How to solve problems with logarithms: examples

Tasks with logarithms Included in the EGE on mathematics for grade 11 in task 5 and task 7, you can find tasks with solutions on our website in the relevant sections. Also, tasks with logarithms are found in the joke of tasks in mathematics. All examples you can find through the search site.

What is logarithm

Logarithms have always been considered a challenging theme in the school course of mathematics. There are many different definitions Logarithm, but most of the textbooks for some reason use the most complex and unsuccessful of them.

We will define the logarithm simply and clearly. To do this, make a table:

So, before us deducts.

Logarithms - properties, formulas, how to solve

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:

based A on the X argument is a 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 on a given base is called. 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 misunderstandings, just 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.

How to count logarithm

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:

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.
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:

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;
Solve relative to the variable B Equation: X \u003d A B;
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

Present the basis and argument as the degree of five: 5 \u003d 5 1; 25 \u003d 5 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;

Received the answer: 2.

A task. Calculate logarithm:

A task. Calculate Logarithm: LOG 4 64

Imagine the basis and argument as a degree of twos: 4 \u003d 2 2; 64 \u003d 2 6;
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;
Received the answer: 3.

A task. Calculate logarithm: log 16 1

Imagine the basis and argument as a degree of two: 16 \u003d 2 4; 1 \u003d 2 0;
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;
Received the answer: 0.

A task. Calculate Logarithm: Log 7 14

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 ;
From the previous point it follows that logarithm is not considered;
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. If there are at least two different factor in the decomposition, the 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;

We also note that the simple numbers themselves are always accurate degrees themselves.

Decimal logarithm

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

from the X argument is a logarithm based on the base 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.

from the X argument 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? This is an irrational number, its exact value Find and write it impossible. 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 logarithov Fair all the rules that are true for ordinary logarithms.

Logarithm. Properties of logarithm (logarithm degree).

How to submit a number in the form of a logarithm?

We use the definition of logarithm.

Logarithm is an indicator of the degree in which the base must be taken to get the number under the sign of the logarithm.

Thus, to represent a certain number C in the form of a logarithm based on a, it is necessary to put a degree with the same base under the logarithm sign as the base of the logarithm, and in terms of the degree of record this number C:

In the form of logarithm you can imagine any number - positive, negative, integer, fractional, rational, irrational:

So that in the stressful conditions of the control or exam does not confuse A and C, you can use such a memorization rule:

what is downstairs goes down, that at the top, go up.

For example, you need to submit a number 2 as a logarithm based on the base 3.

We have two numbers - 2 and 3. These numbers are the foundation and indicator of the degree that we write under the sign of the logarithm. It remains to determine which from these numbers needs to be written down, to the base of the degree, and which is up, in the indicator.

The base 3 in the logarithm record is under the bottom, it means that when we represent the two in the form of a logarithm based on the base 3, 3 also write down to the base.

2 stands above the triple. And in the degree of decend, we write down the top three, that is, in terms of the degree:

Logarithmia. First level.

Logarithmia

Logarithm a positive number b. Based on a.where a\u003e 0, A ≠ 1, is the indicator of the degree in which the number must be issued a., To obtain b..

Definition of logarithm You can briefly record like this:

This equality is fair when b\u003e 0, a\u003e 0, a ≠ 1. It is usually called logarithmic identity.
The locality of the logarithm is called logarithming.

Logarov properties:

Logarithm works:

Logarithm private from division:

Replacing the base of logarithm:

Logarithm:

Logarithm root:

Logarithm with a power base:

Decimal and natural logarithms.

Decimal logarithm Numbers are called the logarithm of this number for the base 10 and write & NBSP LG b.
Natural logarithm numbers call the logarithm of this number based e.where e. - an irrational number, approximately 2.7. At the same time they write ln b..

Other algebra and geometry notes

The main properties of logarithm

Logarithms, like any numbers, can be folded, deduct and convert. But since logarithms are not quite ordinary numbers, there are its own rules that are called basic properties.

These rules must necessarily know - no serious logarithmic task is solved without them. In addition, they are quite a bit - everything can be learned in one day. So, proceed.

Addition and subtraction of logarithms

Consider two logarithms with the same bases: Log A X and Log A Y. Then they can be folded and deducted, and:

log a x + log a y \u003d log a (x · y);
log A x - log a y \u003d log a (x: y).

So, the amount of logarithms is equal to the logarithm of the work, and the difference is the logarithm of private. Please note: the key point here is same grounds. If the foundations are different, these rules do not work!

These formulas will help calculate the logarithmic expression even when individual parts are not considered (see the lesson "What is logarithm"). Take a look at the examples - and make sure:

Log 6 4 + Log 6 9.

Since the bases in logarithms are the same, we use the sum of the sum:
log 6 4 + Log 6 9 \u003d Log 6 (4 · 9) \u003d log 6 36 \u003d 2.

A task. Find the value of the expression: Log 2 48 - Log 2 3.

The foundations are the same, using the difference formula:
log 2 48 - Log 2 3 \u003d Log 2 (48: 3) \u003d log 2 16 \u003d 4.

A task. Find the value of the expression: Log 3 135 - Log 3 5.

Again the foundations are the same, so we have:
log 3 135 - Log 3 5 \u003d log 3 (135: 5) \u003d log 3 27 \u003d 3.

As you can see, the initial expressions are made up of "bad" logarithms, which are not separately considered separately. But after transformation, quite normal numbers are obtained. Many are built on this fact. test papers. But what is the control - such expressions are in full (sometimes - almost unchanged) are offered on the exam.

Executive degree from logarithm

Now a little complicate the task. What if at the base or argument of logarithm costs a degree? Then the indicator of this extent can be taken out of the logarithm sign according to the following rules:

It is easy to see that the last rule follows their first two. But it is better to remember it, in some cases it will significantly reduce the amount of calculations.

Of course, all these rules make sense when compliance with the OTZ Logarithm: A\u003e 0, a ≠ 1, x\u003e 0. And more: learn to apply all formulas not only from left to right, but on the contrary, i.e. You can make numbers facing the logarithm, to the logarithm itself.

How to solve logarithm

That is most often required.

A task. Find the value of the expression: log 7 49 6.

Get rid of the extent in the argument in the first formula:
log 7 49 6 \u003d 6 · Log 7 49 \u003d 6 · 2 \u003d 12

A task. Find the value of the expression:

Note that in the denominator there is a logarithm, the base and the argument of which are accurate degrees: 16 \u003d 2 4; 49 \u003d 7 2. We have:

I think the latest example requires explanation. Where did the logarithms disappeared? Until the last moment, we only work with the denominator. They presented the basis and argument of a logarithm there in the form of degrees and carried out indicators - received a "three-story" fraction.

Now let's look at the basic fraction. The number in the numerator and the denominator is 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 was the answer: 2.

Transition to a new base

Speaking about the rules for the addition and subtraction of logarithms, I specifically emphasized that they work only with the same bases. And what if the foundations are different? What if they are not accurate degrees of the same number?

Formulas for the transition to a new base come to the rescue. We formulate them in the form of theorem:

Let Loga A x be given. Then for any number C such that C\u003e 0 and C ≠ 1, the equality is true:

In particular, if you put C \u003d x, we get:

From the second formula it follows that the base and argument of the logarithm can be changed in places, but at the same time the expression "turns over", i.e. Logarithm turns out to be in the denominator.

These formulas are rare in conventional numerical expressions. Assessing how convenient they are, it is possible only when solving logarithmic equations and inequalities.

However, there are tasks that are generally not solved anywhere as a transition to a new base. Consider a couple of such:

A task. Find the value of the expression: Log 5 16 · Log 2 25.

Note that the arguments of both logarithms are accurate degrees. I will summarize: log 5 16 \u003d log 5 2 4 \u003d 4Log 5 2; Log 2 25 \u003d log 2 5 2 \u003d 2Log 2 5;

And now "invert" the second logarithm:

Since the work does not change from the rearrangement of multipliers, we calmly changed the four and a two, and then sorted out with logarithms.

A task. Find the value of the expression: Log 9 100 · LG 3.

The basis and argument of the first logarithm - accurate degrees. We write it and get rid of the indicators:

Now get rid of the decimal logarithm, by turning to the new base:

Basic logarithmic identity

Often, the solution is required to submit a number as a logarithm for a specified base.

In this case, formulas will help us:

In the first case, the number N becomes an indicator of the extent in the argument. The number n can be absolutely any, because it is just a logarithm value.

The second formula is actually a paraphrassed definition. It is called :.

In fact, what will happen if the number B is in such a degree that the number B to this extent gives the number a? Right: It turns out this the same number a. Carefully read this paragraph again - many "hang" on it.

Like the transition formulas to a new base, the main logarithmic identity is sometimes the only possible solution.

A task. Find the value of the expression:

Note that log 25 64 \u003d log 5 8 - just made a square from the base and the argument of the logarithm. Given the rules for multiplication of degrees with the same base, we get:

If someone is not aware, it was a real task of ege 🙂

Logarithmic unit and logarithmic zero

In conclusion, I will give two identities that it is difficult to name the properties - rather, this is the consequence of the definition of logarithm. They are constantly found in tasks and, which is surprising, create problems even for "advanced" students.

log A a \u003d 1 is. Remember times and forever: the logarithm on any base A from the very base is equal to one.
log A 1 \u003d 0 is. The base a may be any sense, but if the argument is a unit - logarithm equal to zero.! Because a 0 \u003d 1 is a direct consequence of the definition.

That's all properties. Be sure to practice apply them in practice! Download the crib at the beginning of the lesson, print it - and solve the tasks.

basic properties.

logax + Logay \u003d Loga (x · y);
logAX - Logay \u003d Loga (X: Y).

same grounds

Log6 4 + Log6 9.

Now a little complicate the task.

Examples of logarithm solutions

What if at the base or argument of logarithm costs a degree? Then the indicator of this extent can be taken out of the logarithm sign according to the following rules:

Of course, all these rules make sense when compliance with the OTZ Logarithm: A\u003e 0, A ≠ 1, X\u003e

A task. Find the value of the expression:

Transition to a new base

Let Logax LogAx. Then for any number C such that C\u003e 0 and C ≠ 1, the equality is true:

A task. Find the value of the expression:

The main properties of logarithm

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.

The exhibitor is 2,718281828 .... To remember the exhibitor, you can explore the rule: the exhibitor is 2.7 and twice the year of birth of Leo Nikolayevich Tolstoy.

The main properties of logarithm

Knowing this rule will know the exact value of the exhibitory, and the date of birth of Lion Tolstoy.

Examples on logarithmia

Prologate expressions

Example 1.
but). x \u003d 10As ^ 2 (A\u003e 0, C\u003e 0).

By properties 3.5 calculate

4. Where .

Example 2. Find X if

Example 3. Let the value of logarithms are set

Calculate log (x) if

The main properties of logarithm

Logarithms, like any numbers, can be folded, deduct and convert. But since logarithms are not quite ordinary numbers, there are its own rules that are called basic properties.

These rules must necessarily know - no serious logarithmic task is solved without them. In addition, they are quite a bit - everything can be learned in one day. So, proceed.

Addition and subtraction of logarithms

Consider two logarithm with the same bases: Logax and Logay. Then they can be folded and deducted, and:

logax + Logay \u003d Loga (x · y);
logAX - Logay \u003d Loga (X: Y).

These formulas will help calculate the logarithmic expression even when individual parts are not considered (see the lesson "What is logarithm"). Take a look at the examples - and make sure:

Since the bases in logarithms are the same, we use the sum of the sum:
log6 4 + Log6 9 \u003d log6 (4 · 9) \u003d log6 36 \u003d 2.

A task. Find the value of the expression: Log2 48 - Log2 3.

The foundations are the same, using the difference formula:
log2 48 - Log2 3 \u003d Log2 (48: 3) \u003d log2 16 \u003d 4.

A task. Find the value of the expression: Log3 135 - Log3 5.

Again the foundations are the same, so we have:
log3 135 - Log3 5 \u003d log3 (135: 5) \u003d log3 27 \u003d 3.

As you can see, the initial expressions are made up of "bad" logarithms, which are not separately considered separately. But after transformation, quite normal numbers are obtained. In this fact, many test work are built. But what is the control - such expressions are in full (sometimes - almost unchanged) are offered on the exam.

Executive degree from logarithm

It is easy to see that the last rule follows their first two. But it is better to remember it, in some cases it will significantly reduce the amount of calculations.

A task. Find the value of the expression: log7 496.

Get rid of the extent in the argument in the first formula:
log7 496 \u003d 6 · Log7 49 \u003d 6 · 2 \u003d 12

A task. Find the value of the expression:

Note that in the denominator there is a logarithm, the base and the argument of which are accurate degrees: 16 \u003d 24; 49 \u003d 72. We have:

I think the latest example requires explanation. Where did the logarithms disappeared? Until the last moment, we only work with the denominator.

Formulas logarithms. Logarithms Examples of solutions.

They presented the basis and argument of a logarithm there in the form of degrees and carried out indicators - received a "three-story" fraction.

Now let's look at the basic fraction. In a numerator and denominator, the same number is: log2 7. Since log2 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 was the answer: 2.

Transition to a new base

Formulas for the transition to a new base come to the rescue. We formulate them in the form of theorem:

Let Logax LogAx. Then for any number C such that C\u003e 0 and C ≠ 1, the equality is true:

In particular, if you put C \u003d x, we get:

These formulas are rare in conventional numerical expressions. Assessing how convenient they are, it is possible only when solving logarithmic equations and inequalities.

However, there are tasks that are generally not solved anywhere as a transition to a new base. Consider a couple of such:

A task. Find the value of the expression: Log5 16 · Log2 25.

Note that the arguments of both logarithms are accurate degrees. Let's take out the indicators: log5 16 \u003d log5 24 \u003d 4Log5 2; Log2 25 \u003d log2 52 \u003d 2log2 5;

And now "invert" the second logarithm:

Since the work does not change from the rearrangement of multipliers, we calmly changed the four and a two, and then sorted out with logarithms.

A task. Find the value of the expression: Log9 100 · LG 3.

The basis and argument of the first logarithm - accurate degrees. We write it and get rid of the indicators:

Now get rid of the decimal logarithm, by turning to the new base:

Basic logarithmic identity

Often, the solution is required to submit a number as a logarithm for a specified base. In this case, formulas will help us:

In the first case, the number N becomes an indicator of the extent in the argument. The number n can be absolutely any, because it is just a logarithm value.

The second formula is actually a paraphrassed definition. It is called :.

Like the transition formulas to a new base, the main logarithmic identity is sometimes the only possible solution.

A task. Find the value of the expression:

Note that log25 64 \u003d log5 8 - just made a square from the base and the logarithm argument. Given the rules for multiplication of degrees with the same base, we get:

If someone is not aware, it was a real task of ege 🙂

Logarithmic unit and logarithmic zero

logaa \u003d 1 is. Remember times and forever: the logarithm on any base A from the very base is equal to one.
loga 1 \u003d 0 is. The base A may be any sense, but if the argument is a unit - logarithm is zero! Because A0 \u003d 1 is a direct consequence of the definition.

That's all properties. Be sure to practice apply them in practice! Download the crib at the beginning of the lesson, print it - and solve the tasks.

The main properties of logarithm

These properties need to know because, on their basis, almost all tasks are solved and examples are associated with logarithms. The remaining exotic properties can be derived by mathematical manipulations with these formulas

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.

In the calculations of the formula of the sum and the difference of logarithms (3.4) are quite common. The rest are somewhat complicated, but in a number of tasks are indispensable to simplify complex expressions and calculate their values.

There are cases of logarithm

One of the common logarithms are such in which the base is smooth ten, exponential or twice.
The logarithm on the basis of ten is customary to call the decimal logarithm and simplifying the LG (X).

From the record it is clear that the foundations in the record are not written. For example

Natural logarithm is a logarithm for which the exhibitor is based on LN (X)).

The exhibitor is 2,718281828 .... To remember the exhibitor, you can explore the rule: the exhibitor is 2.7 and twice the year of birth of Leo Nikolayevich Tolstoy. Knowing this rule will know the exact value of the exhibitory, and the date of birth of Lion Tolstoy.

And one more important logarithm on the base two designate

The derivative of the logarithm function is equal to a unit divided into a variable

Integral or primitive logarithm is determined by addiction

The above material is enough to solve a wide class of tasks associated with logarithms and logarithmation. For the assimilation of the material, I will give only a few common examples from school program and universities.

Examples on logarithmia

Prologate expressions

Example 1.
but). x \u003d 10As ^ 2 (A\u003e 0, C\u003e 0).

By properties 3.5 calculate

2.
By the properties of the difference logarithms have

3.
Using properties 3.5 find

4. Where .

The form of a complex expression using a number of rules is simplified to mind

Finding the values \u200b\u200bof logarithm

Example 2. Find X if

Decision. For calculation, applicable to the last term of the 3rd and 13 properties

We substitute to write and grieve

Since the grounds are equal, then equating expressions

Logarithmia. First level.

Let the value of logarithms

Calculate log (x) if

Solution: Progriform the variable to paint logarithm through the sum of the terms

On this acquaintance with logarithms and their properties just begins. Exercise in calculations, enrich practical skills - the knowledge gained will soon be needed to solve logarithmic equations. After studying the basic methods of solving such equations, we will expand your knowledge for another at least an important topic - Logarithmic inequalities ...

The main properties of logarithm

Logarithms, like any numbers, can be folded, deduct and convert. But since logarithms are not quite ordinary numbers, there are its own rules that are called basic properties.

These rules must necessarily know - no serious logarithmic task is solved without them. In addition, they are quite a bit - everything can be learned in one day. So, proceed.

Addition and subtraction of logarithms

Consider two logarithm with the same bases: Logax and Logay. Then they can be folded and deducted, and:

logax + Logay \u003d Loga (x · y);
logAX - Logay \u003d Loga (X: Y).

These formulas will help calculate the logarithmic expression even when individual parts are not considered (see the lesson "What is logarithm"). Take a look at the examples - and make sure:

A task. Find the value of the expression: Log6 4 + Log6 9.

Since the bases in logarithms are the same, we use the sum of the sum:
log6 4 + Log6 9 \u003d log6 (4 · 9) \u003d log6 36 \u003d 2.

A task. Find the value of the expression: Log2 48 - Log2 3.

The foundations are the same, using the difference formula:
log2 48 - Log2 3 \u003d Log2 (48: 3) \u003d log2 16 \u003d 4.

A task. Find the value of the expression: Log3 135 - Log3 5.

Again the foundations are the same, so we have:
log3 135 - Log3 5 \u003d log3 (135: 5) \u003d log3 27 \u003d 3.

As you can see, the initial expressions are made up of "bad" logarithms, which are not separately considered separately. But after transformation, quite normal numbers are obtained. In this fact, many test work are built. But what is the control - such expressions are in full (sometimes - almost unchanged) are offered on the exam.

Executive degree from logarithm

It is easy to see that the last rule follows their first two. But it is better to remember it, in some cases it will significantly reduce the amount of calculations.

How to solve logarithm

That is most often required.

A task. Find the value of the expression: log7 496.

Get rid of the extent in the argument in the first formula:
log7 496 \u003d 6 · Log7 49 \u003d 6 · 2 \u003d 12

A task. Find the value of the expression:

Note that in the denominator there is a logarithm, the base and the argument of which are accurate degrees: 16 \u003d 24; 49 \u003d 72. We have:

Transition to a new base

Formulas for the transition to a new base come to the rescue. We formulate them in the form of theorem:

Let Logax LogAx. Then for any number C such that C\u003e 0 and C ≠ 1, the equality is true:

In particular, if you put C \u003d x, we get:

These formulas are rare in conventional numerical expressions. Assessing how convenient they are, it is possible only when solving logarithmic equations and inequalities.

However, there are tasks that are generally not solved anywhere as a transition to a new base. Consider a couple of such:

A task. Find the value of the expression: Log5 16 · Log2 25.

Note that the arguments of both logarithms are accurate degrees. Let's take out the indicators: log5 16 \u003d log5 24 \u003d 4Log5 2; Log2 25 \u003d log2 52 \u003d 2log2 5;

And now "invert" the second logarithm:

Since the work does not change from the rearrangement of multipliers, we calmly changed the four and a two, and then sorted out with logarithms.

A task. Find the value of the expression: Log9 100 · LG 3.

The basis and argument of the first logarithm - accurate degrees. We write it and get rid of the indicators:

Now get rid of the decimal logarithm, by turning to the new base:

Basic logarithmic identity

Often, the solution is required to submit a number as a logarithm for a specified base. In this case, formulas will help us:

In the first case, the number N becomes an indicator of the extent in the argument. The number n can be absolutely any, because it is just a logarithm value.

The second formula is actually a paraphrassed definition. It is called :.

Like the transition formulas to a new base, the main logarithmic identity is sometimes the only possible solution.

A task. Find the value of the expression:

Note that log25 64 \u003d log5 8 - just made a square from the base and the logarithm argument. Given the rules for multiplication of degrees with the same base, we get:

If someone is not aware, it was a real task of ege 🙂

Logarithmic unit and logarithmic zero

logaa \u003d 1 is. Remember times and forever: the logarithm on any base A from the very base is equal to one.
loga 1 \u003d 0 is. The base A may be any sense, but if the argument is a unit - logarithm is zero! Because A0 \u003d 1 is a direct consequence of the definition.

That's all properties. Be sure to practice apply them in practice! Download the crib at the beginning of the lesson, print it - and solve the tasks.

What is logarithm?

Attention!
This topic has additional
Materials in a special section 555.
For those who are strongly "not very ..."
And for those who are "very ...")

What is logarithm? How to solve logarithms? These issues of many graduates are introduced into a stupor. Traditionally, the theme of logarithms is considered complex, incomprehensible and terrible. Especially - equations with logarithms.

It is absolutely wrong. Absolutely! Do not believe? Okay. Now for some 10 - 20 minutes you:

1. Catch what is logarithm.

2. Learn to solve the whole class indicatory equations. Even if nothing heard about them.

3. Learn to calculate simple logarithms.

And for this you will need to know only the multiplication table, but how the number is erected into a degree ...

I feel in doubt ... well, okay, set the time! Go!

To begin with, solve in mind here is such an equation:

If you like this site ...

By the way, I have another couple of interesting sites for you.)

It can be accessed in solving examples and find out your level. Testing with instant check. Learn - with interest!)

You can get acquainted with features and derivatives.

As society developed, mathematics developed complications. Movement from simple to complex. From ordinary accounting by the method of addition and subtraction, with their repeated repetition, they came to the concept of multiplication and division. Reducing a multiply repeated multiplication operation has become the concept of exercise into the degree. The first tables of the dependences of the numbers from the base and the number of construction were drawn up in the VIII century, the Indian mathematician was varamen. With them, you can count the time of logarithms.

Historical essay

The revival of Europe in the XVI century stimulated the development of mechanics. T. i guided a large amount of calculationassociated with multiplication and division of multivalued numbers. The ancient tables provided a big service. They allowed to replace complex operations on simpler - addition and subtraction. The work of Mathematics Michael Stiffel, published in 1544, in which he implemented the idea of \u200b\u200bmany mathematicians was a big step. What made it possible to use tables not only for degrees in the form of simple numbers, but also for arbitrary rational.

In 1614, Scotlandz John Never, developing these ideas, first introduced the new term "logarithm number". New complex tables were drawn up to calculate the logarithms of sinuses and cosines, as well as tangents. It strongly reduced the work of astronomers.

New tables began to appear, which were successfully used by scientists throughout the three centuries. A lot of time passed before the new surgery in algebra acquired its finished look. The definition of logarithm was given, and its properties were studied.

Only in the 20th century, with the advent of the calculator and computer, humanity refused ancient tables that have successfully worked for the XIII centuries.

Today we call the logarithm B based on a number x, which is the degree of number A, so that the number b is. As a formula, this is written: x \u003d log a (b).

For example, Log 3 (9) will be equal to 2. This is obvious if you follow the definition. If 3 is erected into degree 2, then we get 9.

Thus, the defined definition puts only one limitation, the number A and B must be real.

Varieties of logarithm

The classical definition is called the real logarithm and is actually a solution of equation a x \u003d b. Embodiment A \u003d 1 is border and does not represent interest. Attention: 1 to either degree equal to 1.

The real value of logarithm It is determined only at the base and argument greater than 0, while the base should not be equal to 1.

Special place in the field of mathematics Logarithms play, which will be called depending on the value of their foundation:

Rules and restrictions

The fundamental property of logarithms is a rule: the logarithm of the work is equal to the logarithmic amount. Log ABP \u003d 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 function of a private function is equal to the difference difference.

From the previous two rules, it is easy to see that: Log A (B P) \u003d P * Log A (B).

Among other properties can be allocated:

Comment. No need to make a common mistake - the logarithm of the amount is not equal to the sum of logarithms.

For many centuries, the logarithm work operation was quite a time tool. Mathematics enjoyed famous formula Logarithmic theory of decomposition per polynomial:

ln (1 + x) \u003d x - (x ^ 2) / 2 + (x ^ 3) / 3 - (x ^ 4) / 4 + ... + ((-1) ^ (n + 1)) * (( x ^ n) / n), where n - natural number More than 1, which determines the accuracy of the calculation.

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

Since this method is very laborious and when solving practical problems In fact, it is difficult to use pre-composed tables of logarithms, which significantly accelerated all the work.

In some cases, specially composed graphs of logarithms were used, which gave a smaller accuracy, but significantly accelerated the search the desired meaning. The curve of the function y \u003d log a (x), built by several points, allows us to find the function values \u200b\u200bat any other point using a conventional line. Engineers for a long time for these purposes used the so-called millimeter paper.

In the XVII century, the first auxiliary analog computing conditions appeared, which XIX century acquired a finished look. The most successful device was called logarithmic ruler. With all the simplicity of the device, its appearance significantly accelerated the process of all engineering calculations, and it is difficult to overestimate. Currently, few people are familiar with this device.

The appearance of calculators and computers made senseless use of any other devices.

Equations and inequalities

For solutions various equations And the inequalities using logarithms are applied by the following formulas:

Transition from one base to another: LoG A (b) \u003d log c (b) / log c (a);
Consequently previous option: LOG A (B) \u003d 1 / Log B (A).

To solve inequalities it is useful to know:

The logarithm value will be positive only if the base and argument are simultaneously more or less than one; If at least one condition is violated, the logarithm value will be negative.
If the logarithm function applies to the right and left part of the inequality, and the base of the logarithm is greater, then the sign of inequality is preserved; Otherwise, it changes.

Examples of tasks

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

Consider the location of the logarithm to the degree:

Task 3. Calculate 25 ^ log 5 (3). Solution: Under the problems of the task, the record is similar to the following (5 ^ 2) ^ log5 (3) or 5 ^ (2 * log 5 (3)). We write differently: 5 ^ log 5 (3 * 2), or the square of the number as an argument of the function 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 obtain 9.

Practical use

Being an exceptionally mathematical instrument, it seems far from real lifethat logarithm suddenly acquired great importance to describe objects real Mira. It is difficult to find science where it does not apply. This fully applies not only to natural, but also humanitarian areas of knowledge.

Logarithmic dependencies

Here are some examples of numerical dependencies:

Mechanics and physics

Historically, mechanics and physics have always developed using mathematical methods Research 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 by the language of mathematics. We only give the description of the description of physical laws using logarithm.

It is possible to solve the problem of calculating such a complex value as the rocket speed, using the Tsiolkovsky formula, which laid the beginning of the theory of cosmos development:

V \u003d i * ln (m1 / m2), where

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

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

S \u003d k * ln (ω), where

S is a thermodynamic property.
k - Boltzmann's constant.
Ω - statistical weight of different states.

Chemistry

The formulas in chemistry containing the relationship of logarithms will be less obvious. We also give only two examples:

The Nernsta equation, the condition of the oxidation and reductive potential of the medium in relation to the activity of substances and the equilibrium constant.
The calculation of such constants, as an indicator of autoprolization and the acidity of the solution, is also not bypass without our function.

Psychology and biology

And it is completely incomprehensible what is the psychology here. It turns out that the power of the sensation is well described by this function as the reverse ratio of the intensity value of the stimulus to the lower value of the intensity.

After the above examples, it is no longer surprised that the topic of logarithms is widely used in biology. Biological forms corresponding to logarithmic spirals, you can write whole volumes.

Other areas

It seems impossible to exist the world without communication with this function, and it rules all the laws. Especially when the laws of nature are associated with geometric progress. It is worth contacting the site of Matprochi, and such examples there are many in the following areas of activity:

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

With equations we are all familiar with primary classes. Even there we learned to solve the simplest examples, and we must admit that they find their use even in higher Mathematics. With equations, everything is simple, including and square. If you have problems with this topic, we strongly recommend that you repeat it.

Logarithms you probably have also passed. Nevertheless, we consider it important to tell that it is for those who still do not know. The logarithm is equal to the extent to which the base should be taken to make the number that stands to the right of the logarithm sign. Let us give an example on the basis of which you all become clear.

If you are erected 3 into the fourth degree, it turns out 81. Now substitute by analogy, and you will understand the logarithms are completely solved. Now it remains only to combine the two concepts discussed. Initially, the situation seems extremely difficult, but at a closer examination, the weight becomes in its place. We are confident that after this short article you will have no problems in this part of the exam.

Today there are many ways to solve such structures. We will talk about the simplest, efficient and most applicable whes. The solution of logarithmic equations should begin with simple example. The simplest logarithmic equations consist of a function and one variable in it.

It is important to consider that X is inside the argument. A and B must be numbers. In this case, you can simply express the function in a number to the degree. It looks like this.

Of course, the solution of the logarithmic equation This method will lead you to the right answer. The problem of the overwhelming majority of students in this case is that they do not understand what and where it takes. As a result, you have to put up with errors and not to receive the desired points. The most offensive error will be if you confuse letters in places. To solve the equation in this way, you need to get this standard school formula, because it is difficult to understand it.

To make it easier, you can resort to another method - canonical form. The idea is extremely simple. Remote the task again. Remember that the letter A is a number, and not a function or variable. A is not equal to one and more zero. There are no restrictions on b. Now I remember one formula. B can be expressed as follows.

It follows from this that all source equations with logarithms can be represented as:

Now we can discard logarithms. Whenever simple designwhich we have already seen earlier.

The convenience of this formula is that it can be applied in a wide variety of cases, and not just for the most simple designs.

Do not worry about the OOO!

Many experienced mathematicians will notice that we did not pay attention to the definition area. The rule is reduced to the fact that F (x) is necessarily more than 0. No, we did not miss this moment. Now we are talking about another major advantage of the canonical form.

There will be no extra roots here. If the variable will occur only in one place, the definition area is not a necessity. It is performed automatically. To make sure this judgment, make a solution to several simple examples.

How to solve logarithmic equations with different bases

These are already complex logarithmic equations, and the approach to their solution should be special. It is rarely obtained by the notorious canonical form. Let's start our detailed story. We have the following design.

Pay attention to the fraction. There is a logarithm in it. If you see this in the task, it is worth remembering one interesting technique.

What does it mean? Each logarithm can be represented as a private two logarithms with a convenient base. And this formula has a special case that is applicable to this example (we mean if C \u003d b).

It is such a fraction that we see in our example. In this way.

In fact, turned over the fraction and got a more convenient expression. Remember this algorithm!

Now it is necessary that the logarithmic equation did not contain different grounds. Representation by the fraquence.

In mathematics there is a rule based on which one can make a degree from the base. The following construction is obtained.

It would seem that now interferes now to turn our expression in the canonical form and elementary to solve it? Not so simple. We should not be fractions before the logarithm. Correct this situation! The fraction is allowed to endure as an extent.

Respectively.

If the bases are the same, we can remove logarithms and equate the expressions themselves. So the situation will be very easier than it was. The elementary equation will remain, which each of us knew how to decide in 8 or even in grade 7. Calculations you can produce yourself.

We received the only true root of this logarithmic equation. Examples of the solution of the logarithmic equation are quite simple, right? Now and you will get yourself to figure out even with the most complex tasks For the preparation and delivery of the USE.

What is the result?

In the case of any logarithmic equations, we proceed from one very important rule. It is necessary to act so as to bring the expression to the maximum simplicity. In this case, you will have more chances not to simply solve the task correctly, but also make it the most simple and logical way. That is how mathematics always act.

We strongly recommend that you look for complex ways, especially in this case. Remember several simple ruleswhich will allow transform any expression. For example, bring two or three logarithms to one base or withdraw a degree from the ground and win on it.

It is also worth remembering that in solving logarithmic equations it is necessary to constantly train. Gradually, you will go to all more complex designsAnd this will lead you to a confident solution of all options for tasks on the exam. Get ready for exams in advance, and good luck to you!