Logarithms with the same exponents. Natural logarithm, ln x function

Logarithm of b (b > 0) to base a (a > 0, a ≠ 1) is the exponent to which you need to raise the number a to get b.

The base 10 logarithm of b can be written as log(b), and the logarithm to the base e (natural logarithm) - ln(b).

Often used when solving problems with logarithms:

Properties of logarithms

There are four main properties of logarithms.

Let a > 0, a ≠ 1, x > 0 and y > 0.

Property 1. Logarithm of the product

Logarithm of the product is equal to the sum of logarithms:

log a (x ⋅ y) = log a x + log a y

Property 2. Logarithm of the quotient

Logarithm of the quotient is equal to the difference of logarithms:

log a (x / y) = log a x – log a y

Property 3. Logarithm of the degree

Degree logarithm is equal to the product of the degree and the logarithm:

If the base of the logarithm is in the exponent, then another formula applies:

Property 4. Logarithm of the root

This property can be obtained from the property of the logarithm of the degree, since the root of the nth degree is equal to the power of 1/n:

The formula for going from a logarithm in one base to a logarithm in another base

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

Special case:

Comparison of logarithms (inequalities)

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

To compare them, you first need to look at the base of the logarithms a:

If a > 0, then f(x) > g(x) > 0
If 0< a < 1, то 0 < f(x) < g(x)

How to solve problems with logarithms: examples

Tasks with logarithms included in the USE in mathematics for grade 11 in task 5 and task 7, you can find tasks with solutions on our website in the appropriate sections. Also, tasks with logarithms are found in the bank of tasks in mathematics. You can find all examples by searching the site.

What is a logarithm

Logarithms have always been considered a difficult topic in the school mathematics course. There are many different definitions of the logarithm, but for some reason most textbooks use the most complex and unfortunate of them.

We will define the logarithm simply and clearly. Let's create a table for this:

So, we have powers of two.

Logarithms - properties, formulas, how to solve

If you take the number from the bottom line, then you can easily find the power to which you have to raise a 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 - in fact, the definition of the logarithm:

base a of the argument x is the power to which the number a must be raised to get the number x.

Notation: log a x \u003d b, where a is the base, x is the argument, b is actually what the logarithm is equal to.

For example, 2 3 = 8 ⇒ log 2 8 = 3 (the base 2 logarithm of 8 is three because 2 3 = 8). Might as well log 2 64 = 6, because 2 6 = 64.

The operation of finding the logarithm of a number to a given base is called. So let's add a new row 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 considered 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 segment. 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 indefinitely, and they never repeat. If the logarithm turns out to be irrational, it is better to leave it like this: log 2 5, log 3 8, log 5 100.

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

Before us is nothing more than the definition of the logarithm. Remember: the logarithm is the power, to which you need to raise the base to get the argument. It is the base that is raised to a power - in the picture it is highlighted in red. It turns out that the base is always at the bottom! I tell this wonderful rule to my students at the very first lesson - and there is no confusion.

How to count logarithms

We figured out the definition - it remains 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 base must always be greater than zero. This follows from the definition of the degree by a rational exponent, to which the definition of the logarithm is reduced.
The base must be different from unity, since a unit to any power is still a unit. 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 valid range(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) is not imposed. For example, the logarithm may well be negative: log 2 0.5 = −1, because 0.5 = 2 −1 .

However, now we are considering only numerical expressions, where it is not required to know the ODZ of the logarithm. All restrictions have already been taken into account by the compilers of the problems. But when logarithmic equations and inequalities come into play, the DHS requirements will become mandatory. Indeed, in the basis and argument there can be very strong constructions, which do not necessarily correspond to the above restrictions.

Now consider the general scheme for calculating logarithms. It consists of three steps:

Express the base a and the argument x as a power with the smallest possible base greater than one. Along the way, it is better to get rid of decimal fractions;
Solve the equation for the 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 seen already at the first step. The requirement that the base be greater than one is very relevant: this reduces the likelihood of error and greatly simplifies calculations. Similarly with decimal fractions: if you immediately convert them to ordinary ones, there will be many times less errors.

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

A task. Calculate the logarithm: log 5 25

Let's represent the base and the argument as a power of five: 5 = 5 1 ; 25 = 52;

Let's make and solve the equation:
log 5 25 = b ⇒(5 1) b = 5 2 ⇒5 b = 5 2 ⇒ b = 2;

Received an answer: 2.

A task. Calculate the logarithm:

A task. Calculate the logarithm: log 4 64

Let's represent the base and the argument as a power of two: 4 = 2 2 ; 64 = 26;
Let's make and solve the equation:
log 4 64 = b ⇒(2 2) b = 2 6 ⇒2 2b = 2 6 ⇒2b = 6 ⇒ b = 3;
Received an answer: 3.

A task. Calculate the logarithm: log 16 1

Let's represent the base and the argument as a power of two: 16 = 2 4 ; 1 = 20;
Let's make and solve the equation:
log 16 1 = b ⇒(2 4) b = 2 0 ⇒2 4b = 2 0 ⇒4b = 0 ⇒ b = 0;
Received a response: 0.

A task. Calculate the logarithm: log 7 14

Let's represent the base and the argument as a power of seven: 7 = 7 1 ; 14 is not represented as a power of seven, because 7 1< 14 < 7 2 ;
It follows from the previous paragraph that the logarithm is not considered;
The answer is no change: log 7 14.

A small note on the last example. How to make sure that a number is not an exact power of another number? Very simple - just decompose it into prime factors. If there are at least two distinct factors in the expansion, the number is not an exact power.

A task. Find out if the exact powers of the number are: 8; 48; 81; 35; fourteen.

8 \u003d 2 2 2 \u003d 2 3 - the exact degree, because there is only one multiplier;
48 = 6 8 = 3 2 2 2 2 = 3 2 4 is not an exact power because there are two factors: 3 and 2;
81 \u003d 9 9 \u003d 3 3 3 3 \u003d 3 4 - exact degree;
35 = 7 5 - again not an exact degree;
14 \u003d 7 2 - again not an exact degree;

Note also that the prime numbers themselves are always exact powers of themselves.

Decimal logarithm

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

of the x argument is the base 10 logarithm, i.e. the power to which 10 must be raised to obtain x. Designation: lgx.

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

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

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

natural logarithm

There is another logarithm that has its own notation. In a sense, it is even more important than decimal. This is the natural logarithm.

of the x argument is the logarithm to the base e, i.e. the power to which the number e must be raised to get the number x. Designation: lnx.

Many will ask: what is the number e? This is an irrational number, its exact value cannot be found and written down. Here are just the first numbers:
e = 2.718281828459…

We will not delve into 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; log 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 is irrational. Except, of course, unity: ln 1 = 0.

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

Logarithm. Properties of the logarithm (power of the logarithm).

How to represent a number as a logarithm?

We use the definition of a logarithm.

The logarithm is an indicator of the power to which the base must be raised to get the number under the sign of the logarithm.

Thus, in order to represent a certain number c as a logarithm to the base a, it is necessary to put a degree under the sign of the logarithm with the same base as the base of the logarithm, and write this number c into the exponent:

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

In order not to confuse a and c in stressful conditions of a test or exam, you can use the following rule to remember:

what is below goes down, what is above goes up.

For example, you want to represent the number 2 as a logarithm to base 3.

We have two numbers - 2 and 3. These numbers are the base and the exponent, which we will write under the sign of the logarithm. It remains to determine which of these numbers should be written down, in the base of the degree, and which - up, in the exponent.

The base 3 in the record of the logarithm is at the bottom, which means that when we represent the deuce as a logarithm to the base of 3, we will also write 3 down to the base.

2 is higher than 3. And in the notation of the degree, we write the two above the three, that is, in the exponent:

Logarithms. First level.

Logarithms

logarithm positive number b by reason a, where a > 0, a ≠ 1, is the exponent to which the number must be raised. a, To obtain b.

Definition of logarithm can be briefly written like this:

This equality is valid for b > 0, a > 0, a ≠ 1. He is usually called logarithmic identity.
The action of finding the logarithm of a number is called logarithm.

Properties of logarithms:

The logarithm of the product:

Logarithm of the quotient from division:

Replacing the base of the logarithm:

Degree logarithm:

root logarithm:

Logarithm with power base:

Decimal and natural logarithms.

Decimal logarithm numbers call the base 10 logarithm of that number and write lg b
natural logarithm numbers call the logarithm of this number to the base e, where e is an irrational number, approximately equal to 2.7. At the same time, they write ln b.

Other Notes on Algebra and Geometry

Basic properties of logarithms

Logarithms, like any number, can be added, subtracted and converted in every possible way. But since logarithms are not quite ordinary numbers, there are rules here, which are called basic properties.

These rules must be known - no serious logarithmic problem can be solved without them. In addition, there are very few of them - everything can be learned in one day. So let's get started.

Addition and subtraction of logarithms

Consider two logarithms with the same base: log a x and log a y. Then they can be added and subtracted, and:

log a x + log a y = log a (x y);
log a x - log a y = log a (x: y).

So, the sum of the logarithms is equal to the logarithm of the product, and the difference is the logarithm of the quotient. Please note: the key point here is - same grounds. If the bases are different, these rules do not work!

These formulas will help calculate the 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:

log 6 4 + log 6 9.

Since the bases of logarithms are the same, we use the sum formula:
log 6 4 + log 6 9 = log 6 (4 9) = log 6 36 = 2.

A 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.

A 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 considered separately. But after transformations quite normal numbers turn out. Many tests are based on this fact. Yes, control - similar expressions in all seriousness (sometimes - with virtually no changes) are offered at the exam.

Removing the exponent from the logarithm

Now let's complicate the task a little. What if there is a degree in the base or argument of the logarithm? 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 their 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 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 sign of the logarithm into the logarithm itself.

How to solve logarithms

This is what is most often required.

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

Let's get rid of the degree in the argument according to the first formula:
log 7 49 6 = 6 log 7 49 = 6 2 = 12

A task. Find the value of the expression:

Note that the denominator is a logarithm whose base and argument are exact powers: 16 = 2 4 ; 49 = 72. We have:

I think the last example needs clarification. Where have logarithms gone? Until the very last moment, we work only with the denominator. They presented the base and the argument of the logarithm standing there in the form of degrees and took out the indicators - they got a “three-story” fraction.

Now let's look at the main fraction. The numerator and denominator have 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 is 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 bases are different? What if they are not exact powers of the same number?

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

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

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

It follows from the second formula that it is possible to interchange the base and the argument of the logarithm, but in this case the whole expression is “turned over”, i.e. the logarithm is 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 tasks that cannot be solved at all except by moving to a new foundation. Let's consider a couple of these:

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

Note that the arguments of both logarithms are exact exponents. 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 flip the second logarithm:

Since the product does not change from permutation of factors, we calmly multiplied four and two, and then figured out the logarithms.

A 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 it 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 process of solving it is required to represent a number as a logarithm to a given base.

In this case, the 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's just the value of the logarithm.

The second formula is actually a paraphrased definition. It's called like this:

Indeed, what will happen if the number b is raised to such a degree that the number b in this degree gives the number a? That's right: this is the same number a. Read this paragraph carefully again - many people “hang” on it.

Like the new base conversion formulas, the basic logarithmic identity is sometimes the only possible solution.

A task. Find the value of the expression:

Note that log 25 64 = log 5 8 - just took out the square from the base and the argument of the logarithm. Given the rules for multiplying powers with the same base, we get:

If someone is not in the know, this was a real task from the Unified State Examination 🙂

Logarithmic unit and logarithmic zero

In conclusion, I will give two identities that are difficult to call properties - rather, these are consequences from the definition of the logarithm. They are constantly found in problems and, surprisingly, create problems even for "advanced" students.

log a a = 1 is. Remember once and for all: the logarithm to any base a from that base itself is equal to one.
log a 1 = 0 is. The base a can be anything, but if the argument is one, the logarithm is 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.

We continue to study logarithms. In this article we will talk about calculation of logarithms, this process is called logarithm. First, we will deal with the calculation of logarithms by definition. Next, consider how the values of logarithms are found using their properties. After that, we will dwell on the calculation of logarithms through the initially given values of other logarithms. Finally, let's learn how to use tables of logarithms. The whole theory is provided with examples with detailed solutions.

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Computing logarithms by definition

In the simplest cases, it is possible to quickly and easily perform finding the logarithm by definition. Let's take a closer look at how this process takes place.

Its essence is to represent the number b in the form a c , whence, by the definition of the logarithm, the number c is the value of the logarithm. That is, by definition, finding the logarithm corresponds to the following chain of equalities: log a b=log a a c =c .

So, the calculation of the logarithm, by definition, comes down to finding such a number c that a c \u003d b, and the number c itself is the desired value of the logarithm.

Given the information of the previous paragraphs, when the number under the sign of the logarithm is given by some degree of the base of the logarithm, then you can immediately indicate what the logarithm is equal to - it is equal to the exponent. Let's show examples.

Example.

Find log 2 2 −3 , and also calculate the natural logarithm of e 5.3 .

Solution.

The definition of the logarithm allows us to say right away that log 2 2 −3 = −3 . Indeed, the number under the sign of the logarithm is equal to the base 2 to the −3 power.

Similarly, we find the second logarithm: lne 5.3 =5.3.

Answer:

log 2 2 −3 = −3 and lne 5.3 =5.3 .

If the number b under the sign of the logarithm is not given as the power of the base of the logarithm, then you need to carefully consider whether it is possible to come up with a representation of the number b in the form a c . Often this representation is quite obvious, especially when the number under the sign of the logarithm is equal to the base to the power of 1, or 2, or 3, ...

Example.

Compute the logarithms log 5 25 , and .

Solution.

It is easy to see that 25=5 2 , this allows you to calculate the first logarithm: log 5 25=log 5 5 2 =2 .

We proceed to the calculation of the second logarithm. A number can be represented as a power of 7: (see if necessary). Consequently, .

Let's rewrite the third logarithm in the following form. Now you can see that , whence we conclude that . Therefore, by the definition of the logarithm .

Briefly, the solution could be written as follows:

Answer:

log 5 25=2 , And .

When a sufficiently large natural number is under the sign of the logarithm, then it does not hurt to decompose it into prime factors. It often helps to represent such a number as some power of the base of the logarithm, and therefore, to calculate this logarithm by definition.

Example.

Find the value of the logarithm.

Solution.

Some properties of logarithms allow you to immediately specify the value of logarithms. These properties include the property of the logarithm of one and the property of the logarithm of a number equal to the base: log 1 1=log a a 0 =0 and log a a=log a a 1 =1 . That is, when the number 1 or the number a is under the sign of the logarithm, equal to the base of the logarithm, then in these cases the logarithms are 0 and 1, respectively.

Example.

What are the logarithms and lg10 ?

Solution.

Since , it follows from the definition of the logarithm .

In the second example, the number 10 under the sign of the logarithm coincides with its base, so the decimal logarithm of ten is equal to one, that is, lg10=lg10 1 =1 .

Answer:

AND lg10=1 .

Note that computing logarithms by definition (which we discussed in the previous paragraph) implies the use of the equality log a a p =p , which is one of the properties of logarithms.

In practice, when the number under the sign of the logarithm and the base of the logarithm are easily represented as a power of some number, it is very convenient to use the formula , which corresponds to one of the properties of logarithms. Consider an example of finding the logarithm, illustrating the use of this formula.

Example.

Calculate the logarithm of .

Solution.

Answer:

The properties of logarithms not mentioned above are also used in the calculation, but we will talk about this in the following paragraphs.

Finding logarithms in terms of other known logarithms

The information in this paragraph continues the topic of using the properties of logarithms in their calculation. But here the main difference is that the properties of logarithms are used to express the original logarithm in terms of another logarithm, the value of which is known. Let's take an example for clarification. Let's say we know that log 2 3≈1.584963 , then we can find, for example, log 2 6 by doing a little transformation using the properties of the logarithm: log 2 6=log 2 (2 3)=log 2 2+log 2 3≈ 1+1,584963=2,584963 .

In the above example, it was enough for us to use the property of the logarithm of the product. However, much more often you have to use a wider arsenal of properties of logarithms in order to calculate the original logarithm in terms of the given ones.

Example.

Calculate the logarithm of 27 to base 60 if it is known that log 60 2=a and log 60 5=b .

Solution.

So we need to find log 60 27 . It is easy to see that 27=3 3 , and the original logarithm, due to the property of the logarithm of the degree, can be rewritten as 3·log 60 3 .

Now let's see how log 60 3 can be expressed in terms of known logarithms. The property of the logarithm of a number equal to the base allows you to write the equality log 60 60=1 . On the other hand, log 60 60=log60(2 2 3 5)= log 60 2 2 +log 60 3+log 60 5= 2 log 60 2+log 60 3+log 60 5 . In this way, 2 log 60 2+log 60 3+log 60 5=1. Consequently, log 60 3=1−2 log 60 2−log 60 5=1−2 a−b.

Finally, we calculate the original logarithm: log 60 27=3 log 60 3= 3 (1−2 a−b)=3−6 a−3 b.

Answer:

log 60 27=3 (1−2 a−b)=3−6 a−3 b.

Separately, it is worth mentioning the meaning of the formula for the transition to a new base of the logarithm of the form . It allows you to move from logarithms with any base to logarithms with a specific base, the values of which are known or it is possible to find them. Usually, from the original logarithm, according to the transition formula, they switch to logarithms in one of the bases 2, e or 10, since for these bases there are tables of logarithms that allow calculating their values with a certain degree of accuracy. In the next section, we will show how this is done.

Tables of logarithms, their use

For an approximate calculation of the values of the logarithms, one can use logarithm tables. The most commonly used are the base 2 logarithm table, the natural logarithm table, and the decimal logarithm table. When working in the decimal number system, it is convenient to use a table of logarithms to base ten. With its help, we will learn to find the values of logarithms.

The presented table allows, with an accuracy of one ten-thousandth, to find the values of the decimal logarithms of numbers from 1.000 to 9.999 (with three decimal places). We will analyze the principle of finding the value of the logarithm using a table of decimal logarithms using a specific example - it’s clearer. Let's find lg1,256 .

In the left column of the table of decimal logarithms we find the first two digits of the number 1.256, that is, we find 1.2 (this number is circled in blue for clarity). The third digit of the number 1.256 (number 5) is found in the first or last line to the left of the double line (this number is circled in red). The fourth digit of the original number 1.256 (number 6) is found in the first or last line to the right of the double line (this number is circled in green). Now we find the numbers in the cells of the table of logarithms at the intersection of the marked row and the marked columns (these numbers are highlighted in orange). The sum of the marked numbers gives the desired value of the decimal logarithm up to the fourth decimal place, that is, log1.236≈0.0969+0.0021=0.0990.

Is it possible, using the above table, to find the values of the decimal logarithms of numbers that have more than three digits after the decimal point, and also go beyond the limits from 1 to 9.999? Yes, you can. Let's show how this is done with an example.

Let's calculate lg102.76332 . First you need to write number in standard form: 102.76332=1.0276332 10 2 . After that, the mantissa should be rounded up to the third decimal place, we have 1.0276332 10 2 ≈1.028 10 2, while the original decimal logarithm is approximately equal to the logarithm of the resulting number, that is, we take lg102.76332≈lg1.028·10 2 . Now apply the properties of the logarithm: lg1.028 10 2 =lg1.028+lg10 2 =lg1.028+2. Finally, we find the value of the logarithm lg1.028 according to the table of decimal logarithms lg1.028≈0.0086+0.0034=0.012. As a result, the whole process of calculating the logarithm looks like this: lg102.76332=lg1.0276332 10 2 ≈lg1.028 10 2 = lg1.028+lg10 2 =lg1.028+2≈0.012+2=2.012.

In conclusion, it is worth noting that using the table of decimal logarithms, you can calculate the approximate value of any logarithm. To do this, it is enough to use the transition formula to go to decimal logarithms, find their values in the table, and perform the remaining calculations.

For example, let's calculate log 2 3 . According to the formula for the transition to a new base of the logarithm, we have . From the table of decimal logarithms we find lg3≈0.4771 and lg2≈0.3010. In this way, .

Bibliography.

Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the Beginnings of Analysis: A Textbook for Grades 10-11 of General Educational Institutions.
Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).

The basic properties of the natural logarithm, graph, domain of definition, set of values, basic formulas, derivative, integral, expansion in a power series and representation of the function ln x by means of complex numbers are given.

Definition

natural logarithm is the function y = ln x, inverse to the exponent, x \u003d e y , and which is the logarithm to the base of the number e: ln x = log e x.

The natural logarithm is widely used in mathematics because its derivative has the simplest form: (ln x)′ = 1/ x.

Based definitions, the base of the natural logarithm is the number e:
e ≅ 2.718281828459045...;
.

Graph of the function y = ln x.

Graph of the natural logarithm (functions y = ln x) is obtained from the graph of the exponent by mirror reflection about the straight line y = x .

The natural logarithm is defined for positive values of x . It monotonically increases on its domain of definition.

As x → 0 the limit of the natural logarithm is minus infinity ( - ∞ ).

As x → + ∞, the limit of the natural logarithm is plus infinity ( + ∞ ). For large x, the logarithm increases rather slowly. Any power function x a with a positive exponent a grows faster than the logarithm.

Properties of the natural logarithm

Domain of definition, set of values, extrema, increase, decrease

The natural logarithm is a monotonically increasing function, so it has no extrema. The main properties of the natural logarithm are presented in the table.

ln x values

log 1 = 0

Basic formulas for natural logarithms

Formulas arising from the definition of the inverse function:

The main property of logarithms and its consequences

Base replacement formula

Any logarithm can be expressed in terms of natural logarithms using the base change formula:

The proofs of these formulas are presented in the "Logarithm" section.

Inverse function

The reciprocal of the natural logarithm is the exponent.

If , then

If , then .

Derivative ln x

Derivative of the natural logarithm:
.
Derivative of the natural logarithm of the modulo x:
.
Derivative of the nth order:
.
Derivation of formulas > > >

Integral

The integral is calculated by integration by parts:
.
So,

Expressions in terms of complex numbers

Consider a function of a complex variable z :
.
Let's express the complex variable z via module r and argument φ :
.
Using the properties of the logarithm, we have:
.
Or
.
The argument φ is not uniquely defined. If we put
, where n is an integer,
then it will be the same number for different n.

Therefore, the natural logarithm, as a function of a complex variable, is not a single-valued function.

Power series expansion

For , the expansion takes place:

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Higher Educational Institutions, Lan, 2009.

Logarithmic expressions, solution of examples. In this article, we will consider problems related to solving logarithms. The tasks raise the question of finding the value of the expression. It should be noted that the concept of the logarithm is used in many tasks and it is extremely important to understand its meaning. As for the USE, the logarithm is used in solving equations, in applied problems, and also in tasks related to the study of functions.

Here are examples to understand the very meaning of the logarithm:

Basic logarithmic identity:

Properties of logarithms that you must always remember:

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

* * *

* The logarithm of the quotient (fraction) is equal to the difference of the logarithms of the factors.

* * *

* The logarithm of the degree is equal to the product of the exponent and the logarithm of its base.

* * *

*Transition to new base

* * *

More properties:

* * *

Computing logarithms is closely related to using the properties of exponents.

We list some of them:

The essence of this property is that when transferring the numerator to the denominator and vice versa, the sign of the exponent changes to the opposite. For example:

Consequence of this property:

* * *

When raising a power to a power, the base remains the same, but the exponents are multiplied.

* * *

As you can see, the very concept of the logarithm is simple. The main thing is that good practice is needed, which gives a certain skill. Certainly knowledge of formulas is obligatory. If the skill in converting elementary logarithms is not formed, then when solving simple tasks, one can easily make a mistake.

Practice, solve the simplest examples from the math course first, then move on to more complex ones. In the future, I will definitely show how the “ugly” logarithms are solved, there will be no such ones at the exam, but they are of interest, do not miss it!

That's all! Good luck to you!

Sincerely, Alexander Krutitskikh

P.S: I would be grateful if you tell about the site in social networks.

One of the elements of primitive level algebra is the logarithm. The name comes from the Greek language from the word “number” or “degree” and means the degree to which it is necessary to raise the number at the base in order to find the final number.

Types of logarithms

log a b is the logarithm of the number b to the base a (a > 0, a ≠ 1, b > 0);
lg b - decimal logarithm (logarithm base 10, a = 10);
ln b - natural logarithm (logarithm base e, a = e).

How to solve logarithms?

The logarithm of the number b to the base a is an exponent, which requires that the base a be raised to the number b. The result is pronounced like this: “logarithm of b to the base of a”. The solution to logarithmic problems is that you need to determine the given degree by the numbers by the specified numbers. There are some basic rules for determining or solving the logarithm, as well as transforming the notation itself. Using them, logarithmic equations are solved, derivatives are found, integrals are solved, and many other operations are carried out. Basically, the solution to the logarithm itself is its simplified notation. Below are the main formulas and properties:

For any a ; a > 0; a ≠ 1 and for any x ; y > 0.

a log a b = b is the basic logarithmic identity
log a 1 = 0
log a a = 1
log a (x y ) = log a x + log a y
log a x/ y = log a x – log a y
log a 1/x = -log a x
log a x p = p log a x
log a k x = 1/k log a x , for k ≠ 0
log a x = log a c x c
log a x \u003d log b x / log b a - formula for the transition to a new base
log a x = 1/log x a

How to solve logarithms - step by step instructions for solving

First, write down the required equation.

Please note: if the base logarithm is 10, then the record is shortened, a decimal logarithm is obtained. If there is a natural number e, then we write down, reducing to a natural logarithm. It means that the result of all logarithms is the power to which the base number is raised to obtain the number b.

Directly, the solution lies in the calculation of this degree. Before solving an expression with a logarithm, it must be simplified according to the rule, that is, using formulas. You can find the main identities by going back a little in the article.

When adding and subtracting logarithms with two different numbers but with the same base, replace with a single logarithm with the product or division of the numbers b and c, respectively. In this case, you can apply the transition formula to another base (see above).

If you are using expressions to simplify the logarithm, there are some limitations to consider. And that is: the base of the logarithm a is only a positive number, but not equal to one. The number b, like a, must be greater than zero.

There are cases when, having simplified the expression, you will not be able to calculate the logarithm in numerical form. It happens that such an expression does not make sense, because many degrees are irrational numbers. Under this condition, leave the power of the number as a logarithm.