Natural logarithm Exhibitors is equal. Logarithm

often take a digit e. = 2,718281828 . Logarithms on this basis are called natural. When performing computing with natural logarithms, it is generally accepted with a sign l.n., but not log.; At the same time 2,718281828 The determining base does not indicate.

In other words, the wording will look at: natural logarithm numbers h. - This is an indicator in which the number needs to be issued e., To obtain x..

So, ln (7,389 ...)\u003d 2, since e. 2 =7,389... . Natural logarithm of the very number e.\u003d 1, because e. 1 =e., and the natural logarithm unit is zero, because e. 0 = 1.

Name e.determines the limit of monotonous limited sequence

calculated that e. = 2,7182818284... .

Very often for fixing in memory of any number, the numbers of the required number are associated with some outstanding date. Memorization rate of the first nine signs of the number e. After the comma will increase, if noted that 1828 is the year of birth of a lion thick!

To date, there are quite complete tables of natural logarithms.

Schedule of natural logarithm (Functions y \u003d.lN X.) is a consequence of the schedule of the exponent with a mirror reflection relatively direct y \u003d x. and has the form:

Natural logarithm can be found for each positive real number. a. as an area under the curve y. = 1/x. from 1 before a..

The elementality of this wording, which is docked with many other formulas, in which natural logarithm is involved, was the reason for the formation of the name "natural".

If analyzing natural logarithmas a real function of a valid variable, then it appears inverse function To the exponential function, which comes down to identities:

e ln (A) \u003d A (A\u003e 0)

ln (e a) \u003d a

By analogy with all logarithms, natural logarithm converts multiplication into addition, division into subtraction:

lN.(xY.) = lN.(x.) + lN.(y.)

lN.(x / y) \u003d lNX - lNY.

Logarithm can be found for each positive foundation that is not equal to one, and not just for e.But logarithms for other bases differ from the natural logarithm only by a constant factor, and usually determined in terms of natural logarithm.

Analyzed schedule of natural logarithm, We get that it exists with positive values \u200b\u200bof the variable x.. He monotonically increases on its field of definition.

For x. → 0 The limit of natural logarithm is minus infinity ( -∞ ). Dr. x → + ∞ The limit of natural logarithm performs plus infinity ( + ∞ ). With large x.logarithm increases pretty slowly. Any powerful function x A.with a positive indicator a. Retrieves the faster logarithm. Natural logarithm is a monotonously increasing function, so there are no extremes.

Using natural logarithov It is very rational when passing the highest mathematics. Thus, the use of logarithm is convenient to find the response of equations in which unknown persons appear as an indicator. Application in the calculations of natural logarithm makes it possible to pretty alleviate a large number of mathematical formulas. Logarithmia based on e. Present in solving a significant number of physical problems and naturally included in the mathematical description of individual chemical, biological and other processes. Thus, logarithms are used to calculate the constant decay for the known half-life period, or to calculate the decay time in solving radioactivity problems. They act in the lead role in many sections of mathematics and practical sciences, they are resorted to the field of finance to solve a large number of tasks, including in the calculation of complex interest.

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

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

Basic logarithmic identity

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

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

Two obvious consequences of the definition of logarithm

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

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

Logarithm works and logarithm private

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

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

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

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

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

The degree can be made for the logarithm sign

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

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

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

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

Formula of the transition to a new base

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

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

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

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

Some simple examples with logarithms

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

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

Table formulas related to logarithms

A log a b \u003d b (a\u003e 0, a ≠ 1)

Log A A \u003d 1 (A\u003e 0, A ≠ 1)

Log A 1 \u003d 0 (A\u003e 0, A ≠ 1)

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

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

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

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

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

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

And now - actually, the definition of logarithm:

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

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

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

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

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

Unfortunately, not all logarithms are considered so easy. For example, try to find Log 2 5. Numbers 5 are not in the table, but the logic suggests 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 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.

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

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

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

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

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

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

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

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

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

A task. Calculate logarithm: log 5 25

1. Present the basis and argument as the degree of five: 5 \u003d 5 1; 25 \u003d 5 2;

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;

2. Received the answer: 2.

A task. Calculate logarithm:

A task. Calculate Logarithm: LOG 4 64

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

A task. Calculate logarithm: log 16 1

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

A task. Calculate Logarithm: Log 7 14

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

Little remark to the last example. How to make sure that the number is not the exact degree of another number? Very simple - enough to decompose it on simple factors. 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 a special name and designation.

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

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

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

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

Natural logarithm

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

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

Many will ask: what else in the number e? This is an irrational number, its exact value to 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 logarithms, all the rules that are true for ordinary logarithms are valid.

Fig. 16. Behavior of the function f (x) \u003d x4 4x3

When moving through the point x \u003d 0, the derivative does not change the sign: the function decreases both on the interval (1; 0] and on the interval. Therefore, the point x \u003d 0 is a saddle point of the function.

But when switching through the point x \u003d 3, the derivative changes the sign C () to (+). In the interval)