Algebraic operations on logarithms. Natural logarithm, function ln x. Examples for logarithms

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 that?

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 - 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 is, but it cannot be raised to negative at all, since this 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 tinker 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 basis 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:

How do you like the article?

If you are reading these lines, then you have read the entire article.

And that's cool!

Now tell us how do you like the article?

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

Write to us in the comments below.

And, yes, good luck on your exams.

On the Unified State Exam and the Unified State Exam and in life in general

main properties.

logax + logay = loga(x y);
logax − logay = loga (x: y).

identical grounds

Log6 4 + log6 9.

Now let's complicate the task a little.

Examples of solving logarithms

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:

Of course, all these rules make sense if the ODZ of the logarithm is observed: a > 0, a ≠ 1, x >

Task. Find the meaning of the expression:

Transition to a new foundation

Let it be given logarithm log ax. Then for any number c such that c > 0 and c ≠ 1, the equality is true:

Task. Find the meaning of the expression:

Basic properties of the logarithm

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

The exponent is 2.718281828…. To remember the exponent, you can study the rule: the exponent is equal to 2.7 and twice the year of birth of Leo Nikolaevich Tolstoy.

Basic properties of logarithms

Knowing this rule, you will know both the exact value of the exponent and the date of birth of Leo Tolstoy.

Examples for logarithms

Logarithm expressions

Example 1.
A). x=10ac^2 (a>0,c>0).

Using properties 3.5 we calculate

Example 2. Find x if

Example 3. Let the value of logarithms be given

Calculate log(x) if

Basic properties of logarithms

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 - without them, not a single serious logarithmic problem can be solved. 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: logax and logay. Then they can be added and subtracted, and:

logax + logay = loga(x y);
logax − logay = loga (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 counted (see the lesson “What is a logarithm”). Take a look at the examples and see:

Since logarithms have the same bases, we use the sum formula:
log6 4 + log6 9 = log6 (4 9) = log6 36 = 2.

Task. Find the value of the expression: log2 48 − log2 3.

The bases are the same, we use the difference formula:
log2 48 − log2 3 = log2 (48: 3) = log2 16 = 4.

Task. Find the value of the expression: log3 135 − log3 5.

Again the bases are the same, so we have:
log3 135 − log3 5 = log3 (135: 5) = log3 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 test papers. 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

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: log7 496.

Let's get rid of the degree in the argument using the first formula:
log7 496 = 6 log7 49 = 6 2 = 12

Task. Find the meaning of the expression:

Note that the denominator contains a logarithm, the base and argument of which are exact powers: 16 = 24; 49 = 72. We have:

I think to last example clarification required. Where have logarithms gone? Until the very last moment we work only with the denominator.

Logarithm formulas. Logarithms examples solutions.

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: 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 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 logax be given. Then for any number c such that c > 0 and c ≠ 1, the equality is true:

In particular, if we set c = x, we get:

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 ordinary numerical expressions. It is possible to evaluate how convenient they are only when solving 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: log5 16 log2 25.

Note that the arguments of both logarithms contain exact powers. Let's take out the indicators: log5 16 = log5 24 = 4log5 2; log2 25 = log2 52 = 2log2 5;

Now let’s “reverse” the second logarithm:

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: log9 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:

Now let's get rid of the decimal logarithm by moving to a new base:

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 the exponent in the argument. The number n can be absolutely anything, because it is just a logarithm value.

The second formula is actually a paraphrased definition. That's what it's called: .

In fact, what happens if the number b is raised to such a power that the number b to this power gives the number a? That's right: the result is the 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:

Note that log25 64 = log5 8 - 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:

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.

logaa = 1 is. Remember once and for all: the logarithm to any base a of that base itself is equal to one.
loga 1 = 0 is. The base a can be anything, but if the argument contains one, the logarithm is equal to zero! Because a0 = 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.

Basic properties of the logarithm

It is necessary to know the above properties, since almost all problems and examples related to logarithms are solved on their basis. The rest of the exotic properties can be derived through mathematical manipulations with these formulas

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

When calculating the formula for the sum and difference of logarithms (3.4) you come across quite often. The rest are somewhat complex, but in a number of tasks they are indispensable for simplifying complex expressions and calculating their values.

Common cases of logarithms

Some of the common logarithms are those in which the base is even ten, exponential or two.
The logarithm to base ten is usually called the decimal logarithm and is simply denoted by lg(x).

It is clear from the recording that the basics are not written in the recording. For example

Natural logarithm is a logarithm with an exponent as its basis (denoted by ln(x)).

The exponent is 2.718281828…. To remember the exponent, you can study the rule: the exponent is equal to 2.7 and twice the year of birth of Leo Nikolaevich Tolstoy. Knowing this rule, you will know both the exact value of the exponent and the date of birth of Leo Tolstoy.

And another important logarithm to base two is denoted by

The derivative of the logarithm of a function is equal to one divided by the variable

The integral or antiderivative logarithm is determined by the relationship

The given material is enough for you to solve a wide class of problems related to logarithms and logarithms. To help you understand the material, I will give just a few common examples from school curriculum and universities.

Examples for logarithms

Logarithm expressions

Example 1.
A). x=10ac^2 (a>0,c>0).

Using properties 3.5 we calculate

2.
By the property of difference of logarithms we have

3.
Using properties 3.5 we find

By the look complex expression using a number of rules is simplified to form

Finding logarithm values

Example 2. Find x if

Solution. For calculation, we apply to the last term 5 and 13 properties

We put it on record and mourn

Since the bases are equal, we equate the expressions

Logarithms. First level.

Let the value of logarithms be given

Calculate log(x) if

Solution: Let's take a logarithm of the variable to write the logarithm through the sum of its terms

This is just the beginning of our acquaintance with logarithms and their properties. Practice calculations, enrich your practical skills - you will soon need the knowledge you gain to solve logarithmic equations. Having studied the basic methods for solving such equations, we will expand your knowledge for another no less important topic- logarithmic inequalities...

Basic properties of logarithms

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.

Adding and subtracting logarithms

Consider two logarithms with the same bases: logax and logay. Then they can be added and subtracted, and:

logax + logay = loga(x y);
logax − logay = loga (x: y).

These formulas will help you calculate a logarithmic expression even when its individual parts are not considered (see the lesson “What is a logarithm”). Take a look at the examples and see:

Task. Find the value of the expression: log6 4 + log6 9.

Since logarithms have the same bases, we use the sum formula:
log6 4 + log6 9 = log6 (4 9) = log6 36 = 2.

Task. Find the value of the expression: log2 48 − log2 3.

The bases are the same, we use the difference formula:
log2 48 − log2 3 = log2 (48: 3) = log2 16 = 4.

Task. Find the value of the expression: log3 135 − log3 5.

Again the bases are the same, so we have:
log3 135 − log3 5 = log3 (135: 5) = log3 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 tests are based on this fact. 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.

How to solve logarithms

This is what is most often required.

Task. Find the value of the expression: log7 496.

Let's get rid of the degree in the argument using the first formula:
log7 496 = 6 log7 49 = 6 2 = 12

Task. Find the meaning of the expression:

Note that the denominator contains a logarithm, the base and argument of which are exact powers: 16 = 24; 49 = 72. We have:

I think the last example requires some clarification. 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.

Transition to a new foundation

Formulas for transition to a new foundation come to the rescue. Let us formulate them in the form of a theorem:

Let the logarithm logax be given. Then for any number c such that c > 0 and c ≠ 1, the equality is true:

In particular, if we set c = x, we get:

These formulas are rarely found in ordinary numerical expressions. It is possible to evaluate how convenient they are only when solving logarithmic equations and inequalities.

However, there are 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: log5 16 log2 25.

Note that the arguments of both logarithms contain exact powers. Let's take out the indicators: log5 16 = log5 24 = 4log5 2; log2 25 = log2 52 = 2log2 5;

Now let’s “reverse” the second logarithm:

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: log9 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:

Now let's get rid of the decimal logarithm by moving to a new base:

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 the exponent in the argument. The number n can be absolutely anything, because it is just a logarithm value.

The second formula is actually a paraphrased definition. That's what it's called: .

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:

Note that log25 64 = log5 8 - 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:

If anyone doesn’t know, this was a real task from the Unified State Exam :)

Logarithmic unit and logarithmic zero

logaa = 1 is. Remember once and for all: the logarithm to any base a of that base itself is equal to one.
loga 1 = 0 is. The base a can be anything, but if the argument contains one, the logarithm is equal to zero! Because a0 = 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.

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.

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).

These formulas will help you calculate a 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 tests are based on this fact. 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

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]

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

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]

These formulas are rarely found in ordinary numerical expressions. It is possible to evaluate how convenient they are only when solving logarithmic equations and inequalities.

However, there are 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 the decimal logarithm by 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 - 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

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.

Logarithm of a number N based on A called exponent X , to which you need to build A to get the number N

Provided that
,
,

From the definition of logarithm it follows that
, i.e.
- this equality is the basic logarithmic identity.

Logarithms to base 10 are called decimal logarithms. Instead of
write
.

Logarithms to the base e are called natural and are designated
.

Basic properties of logarithms.

The logarithm of one is equal to zero for any base.

The logarithm of the product is equal to the sum of the logarithms of the factors.

3) The logarithm of the quotient is equal to the difference of the logarithms

Factor
called the modulus of transition from logarithms to the base a to logarithms at the base b .

Using properties 2-5, it is often possible to reduce the logarithm of a complex expression to the result of simple arithmetic operations on logarithms.

For example,

Such transformations of a logarithm are called logarithms. Transformations inverse to logarithms are called potentiation.

Chapter 2. Elements of higher mathematics.

1. Limits

Limit of the function
is a finite number A if, as xx 0 for each predetermined
, there is such a number
that as soon as
, That
.

A function that has a limit differs from it by an infinitesimal amount:
, where- b.m.v., i.e.
.

Example. Consider the function
.

When striving
, function y tends to zero:

1.1. Basic theorems about limits.

The limit of a constant value is equal to this constant value

Amount (difference) limit finite number functions is equal to the sum (difference) of the limits of these functions.

The limit of the product of a finite number of functions is equal to the product of the limits of these functions.

The limit of the quotient of two functions is equal to the quotient of the limits of these functions if the limit of the denominator is not zero.

Wonderful Limits

,
, Where

1.2. Limit Calculation Examples

However, not all limits are calculated so easily. More often, calculating the limit comes down to revealing an uncertainty of the type: or .

2. Derivative of a function

Let us have a function
, continuous on the segment
.

Argument got some increase
. Then the function will receive an increment
.

Argument value corresponds to the function value
.

Argument value
corresponds to the function value.

Hence, .

Let us find the limit of this ratio at
. If this limit exists, then it is called the derivative of the given function.

Definition 3 Derivative of a given function
by argument is called the limit of the ratio of the increment of a function to the increment of the argument, when the increment of the argument arbitrarily tends to zero.

Derivative of a function
can be designated as follows:

; ; ; .

Definition 4The operation of finding the derivative of a function is called differentiation.

2.1. Mechanical meaning of derivative.

Let's consider the rectilinear motion of some rigid body or material point.

Let at some point in time moving point
was at a distance from the starting position
.

After some period of time
she moved a distance
. Attitude =- average speed of a material point
. Let us find the limit of this ratio, taking into account that
.

Consequently, determining the instantaneous speed of movement of a material point is reduced to finding the derivative of the path with respect to time.

2.2. Geometric value of the derivative

Let us have a graphically defined function
.

Rice. 1. Geometric meaning of derivative

If
, then point
, will move along the curve, approaching the point
.

Hence
, i.e. the value of the derivative for a given value of the argument numerically equal to the tangent of the angle formed by the tangent at a given point with the positive direction of the axis
.

2.3. Table of basic differentiation formulas.

Power function

Exponential function

Logarithmic function

Trigonometric function

Inverse trigonometric function

2.4. Rules of differentiation.

Derivative of

Derivative of the sum (difference) of functions

Derivative of the product of two functions

Derivative of the quotient of two functions

2.5. Derivative of complex function.

Let the function be given
such that it can be represented in the form

And
, where the variable is an intermediate argument, then

The derivative of a complex function is equal to the product of the derivative of the given function with respect to the intermediate argument and the derivative of the intermediate argument with respect to x.

Example 1.

Example 2.

3. Differential function.

Let there be
, differentiable on some interval
let it go at this function has a derivative

then we can write

(1),

Where - an infinitesimal quantity,

since when

Multiplying all terms of equality (1) by
we have:

Where
- b.m.v. higher order.

Magnitude
called the differential of the function
and is designated

3.1. Geometric value of the differential.

Let the function be given
.

Fig.2. Geometric meaning of differential.

Obviously, the differential of the function
is equal to the increment of the ordinate of the tangent at a given point.

3.2. Derivatives and differentials of various orders.

If there
, Then
is called the first derivative.

The derivative of the first derivative is called the second-order derivative and is written
.

Derivative of the nth order of the function
is called the (n-1)th order derivative and is written:

The differential of the differential of a function is called the second differential or second order differential.

.

.

3.3 Solving biological problems using differentiation.

Task 1. Studies have shown that the growth of a colony of microorganisms obeys the law
, Where N – number of microorganisms (in thousands), t – time (days).

b) Will the population of the colony increase or decrease during this period?

Answer. The size of the colony will increase.

Task 2. The water in the lake is periodically tested to monitor the content of pathogenic bacteria. Through t days after testing, the concentration of bacteria is determined by the ratio

When will the lake have a minimum concentration of bacteria and will it be possible to swim in it?

Solution: A function reaches max or min when its derivative is zero.

Let's determine the max or min will be in 6 days. To do this, let's take the second derivative.

Answer: After 6 days there will be a minimum concentration of bacteria.

So, we have powers of two. If you take the number from the bottom line, you can easily find the power to which you will have to raise two to get this number. For example, to get 16, you need to raise two to the fourth power. And to get 64, you need to raise two to the sixth power. This can be seen from the table.

And now, actually, the definition of the logarithm:

The base a logarithm of x is the power to which a must be raised to get x.

Notation: log a x = b, where a is the base, x is the argument, b is what the logarithm is actually equal to.

For example, 2 3 = 8 ⇒ log 2 8 = 3 (the base 2 logarithm of 8 is three because 2 3 = 8). With the same success, log 2 64 = 6, since 2 6 = 64.

The operation of finding the logarithm of a number to a given base is called logarithmization. So, let's add a new line to our table:

2 1	2 2	2 3	2 4	2 5	2 6
2	4	8	16	32	64
log 2 2 = 1	log 2 4 = 2	log 2 8 = 3	log 2 16 = 4	log 2 32 = 5	log 2 64 = 6

Unfortunately, not all logarithms are calculated so easily. For example, try to find log 2 5. The number 5 is not in the table, but logic dictates that the logarithm will lie somewhere on the interval. Because 2 2< 5 < 2 3 , а чем больше степень двойки, тем больше получится число.

Such numbers are called irrational: the numbers after the decimal point can be written ad infinitum, and they are never repeated. If the logarithm turns out to be irrational, it is better to leave it that way: log 2 5, log 3 8, log 5 100.

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

Before us is nothing more than the definition of a logarithm. Remember: logarithm is a power, into which the base must be built in order to obtain an argument. It is the base that is raised to a power - it is highlighted in red in the picture. It turns out that the base is always at the bottom! I tell my students this wonderful rule at the very first lesson - and no confusion arises.

We've figured out the definition - all that's left is to learn how to count logarithms, i.e. get rid of the "log" sign. To begin with, we note that two important facts follow from the definition:

The argument and the base must always be greater than zero. This follows from the definition of the degree rational indicator, to which the definition of a logarithm comes down.
The base must be different from one, since one to any degree still remains one. Because of this, the question “to what power must one be raised to get two” is meaningless. There is no such degree!

Such restrictions are called range of acceptable values(ODZ). It turns out that the ODZ of the logarithm looks like this: log a x = b ⇒ x > 0, a > 0, a ≠ 1.

Note that there are no restrictions on the number b (the value of the logarithm). For example, the logarithm may well be negative: log 2 0.5 = −1, because 0.5 = 2 −1.

However, now we are only considering numeric expressions, where it is not required to know the logarithm's CVD. All restrictions have already been taken into account by the authors of the problems. But when they go logarithmic equations and inequalities, DHS requirements will become mandatory. After all, the basis and argument may contain very strong constructions that do not necessarily correspond to the above restrictions.

Now let's consider general scheme calculating logarithms. It consists of three steps:

Express the base a and the argument x as a power with the minimum possible base greater than one. Along the way, it’s better to get rid of decimals;
Solve the equation for variable b: x = a b ;
The resulting number b will be the answer.

That's all! If the logarithm turns out to be irrational, this will be visible already in the first step. The requirement that the base be greater than one is very important: this reduces the likelihood of error and greatly simplifies the calculations. Same with decimals: if you immediately convert them to regular ones, there will be many fewer errors.

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

Task. Calculate the logarithm: log 5 25

Let's imagine the base and argument as a power of five: 5 = 5 1 ; 25 = 5 2 ;
Let's create and solve the equation:
log 5 25 = b ⇒ (5 1) b = 5 2 ⇒ 5 b = 5 2 ⇒ b = 2;
We received the answer: 2.

Task. Calculate the logarithm:

Task. Calculate the logarithm: log 4 64

Let's imagine the base and argument as a power of two: 4 = 2 2 ; 64 = 2 6 ;
Let's create and solve the equation:
log 4 64 = b ⇒ (2 2) b = 2 6 ⇒ 2 2b = 2 6 ⇒ 2b = 6 ⇒ b = 3;
We received the answer: 3.

Task. Calculate the logarithm: log 16 1

Let's imagine the base and argument as a power of two: 16 = 2 4 ; 1 = 2 0 ;
Let's create and solve the equation:
log 16 1 = b ⇒ (2 4) b = 2 0 ⇒ 2 4b = 2 0 ⇒ 4b = 0 ⇒ b = 0;
We received the answer: 0.

Task. Calculate the logarithm: log 7 14

Let's imagine the base and argument as a power of seven: 7 = 7 1 ; 14 cannot be represented as a power of seven, since 7 1< 14 < 7 2 ;
From the previous paragraph it follows that the logarithm does not count;
The answer is no change: log 7 14.

A small note on the last example. How can you be sure that a number is not an exact power of another number? It's very simple - just break it down into prime factors. And if such factors cannot be collected into powers with the same exponents, then the original number is not an exact power.

Task. Find out whether the numbers are exact powers: 8; 48; 81; 35; 14.

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

Let us also note that we ourselves prime numbers are always exact degrees of themselves.

Decimal logarithm

Some logarithms are so common that they have a special name and symbol.

The decimal logarithm of x is the logarithm to base 10, i.e. The power to which the number 10 must be raised to obtain the number x. Designation: lg x.

For example, log 10 = 1; lg 100 = 2; lg 1000 = 3 - etc.

From now on, when a phrase like “Find lg 0.01” appears in a textbook, know that this is not a typo. This decimal logarithm. However, if you are unfamiliar with this notation, you can always rewrite it:
log x = log 10 x

Everything that is true for ordinary logarithms is also true for decimal logarithms.

Natural logarithm

There is another logarithm that has its own designation. In some ways, it's even more important than decimal. We are talking about the natural logarithm.

The natural logarithm of x is the logarithm to base e, i.e. the power to which the number e must be raised to obtain the number x. Designation: ln x .

Many will ask: what is the number e? This is an irrational number; its exact value cannot be found and written down. I will give only the first figures:
e = 2.718281828459...

We won’t go into detail about what this number is and why it is needed. Just remember that e is the base of the natural logarithm:
ln x = log e x

Thus ln e = 1; ln e 2 = 2; ln e 16 = 16 - etc. On the other hand, ln 2 is an irrational number. In general, the natural logarithm of any rational number irrational. Except, of course, for one: ln 1 = 0.

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

Algebraic operations on logarithms. Natural logarithm, function ln x. Examples for logarithms

Range of acceptable values ​​(APV) of the logarithm

Basic logarithmic identity

Properties of logarithms

Property 1:

Property 2: Sum of logarithms

Property 3: Difference of logarithms:

Property 4: Taking the exponent out of the logarithm argument:

Property 5: Taking the exponent from the base of the logarithm:

Property 6: Removing the exponent from the base and argument of the logarithm:

Property 7: Transition to a new base:

Property 8: Swap the base and argument of the logarithm:

Examples of solving logarithms

Transition to a new foundation

Basic properties of the logarithm

Basic properties of logarithms

Examples for logarithms

Logarithm expressions

Basic properties of logarithms

Adding and subtracting logarithms

Extracting the exponent from the logarithm

Logarithm formulas. Logarithms examples solutions.

Transition to a new foundation

Basic logarithmic identity

Logarithmic unit and logarithmic zero

Basic properties of the logarithm

Common cases of logarithms

Examples for logarithms

Logarithm expressions

Finding logarithm values

Logarithms. First level.

Basic properties of logarithms

Adding and subtracting logarithms

Extracting the exponent from the logarithm

How to solve logarithms

Transition to a new foundation

Basic logarithmic identity

Logarithmic unit and logarithmic zero

Adding and subtracting logarithms

Extracting the exponent from the logarithm

Transition to a new foundation

Basic logarithmic identity

Logarithmic unit and logarithmic zero

Chapter 2. Elements of higher mathematics.

2.1. Mechanical meaning of derivative.

Decimal logarithm

Natural logarithm

Range of acceptable values (APV) of the logarithm