Properties of logarithms and examples of their solutions. Comprehensive guide (2020). The logarithm of b to base a is the exponent, in Proof of the basic formulas for logarithms

So, we have before us powers of two. If you take the number from the bottom line, then you can easily find the degree to which you have to raise the 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 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.

For example, 2 3 \u003d 8 ⇒ log 2 8 \u003d 3 (logarithm base 2 of 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 a number in a given base is called the logarithm. 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 \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 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 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 that way: 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 are confused about where the foundation 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: logarithm is the degreeto which the base must be raised to get the argument. It is the base that is raised to the 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 no confusion arises.

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:

1. Argument and radix must always be greater than zero. This follows from the definition of the degree by a rational indicator, to which the definition of the logarithm is reduced.
2. The base must be different from one, since one is still one to any degree. Because of this, the question “to what degree one should raise one to get a two” is meaningless. There is no such degree!

Such restrictions are called range of valid values (ODZ). It turns out that the ODZ of the logarithm looks like this: log a x \u003d b ⇒ x\u003e 0, a\u003e 0, a ≠ 1.

Note that there is no restriction on the number b (the value of the logarithm). For example, the 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 knowing the ODV of the logarithm is not required. All restrictions have already been taken into account by the task compilers. But when the logarithmic equations and inequalities come in, the DHS requirements will become mandatory. Indeed, at the base and in the argument there can be very strong constructions that do not necessarily correspond to the above restrictions.

Now let's look at the general scheme for calculating logarithms. It consists of three steps:

1. Represent base a and 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;
2. Solve the equation for variable b: x \u003d a b;
3. 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 for the base to be greater than one is highly relevant: this reduces the probability of error and greatly simplifies calculations. Similarly, with decimal fractions: if you immediately convert them into 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

1. Let's represent the base and the argument as a power of five: 5 \u003d 5 1; 25 \u003d 5 2;
2. Let's compose and solve the equation:
   log 5 25 \u003d b ⇒ (5 1) b \u003d 5 2 ⇒ 5 b \u003d 5 2 ⇒ b \u003d 2;
3. Received the answer: 2.

A task. Calculate the logarithm:

A task. Calculate the logarithm: log 4 64

1. Let's represent the base and the argument as a power of two: 4 \u003d 2 2; 64 \u003d 2 6;
2. Let's compose 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 the logarithm: log 16 1

1. Let's represent the base and the argument as a power of two: 16 \u003d 2 4; 1 \u003d 2 0;
2. Let's compose 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 the log of: log 7 14

1. We represent the base and the argument as a power of seven: 7 \u003d 7 1; 14 is not represented as a power of seven, since 7 1< 14 < 7 2 ;
2. It follows from the previous paragraph that the logarithm is not considered;
3. The answer is no change: log 7 14.

A small note on the last example. How do you ensure that a number is not an exact power of another number? It's very simple - just factor it into prime factors. And if such factors cannot be collected in powers with the same indicators, then the original 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 factor;
48 \u003d 6 · 8 \u003d 3 · 2 · 2 · 2 · 2 \u003d 3 · 2 4 - 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 - exact degree;
35 \u003d 7 · 5 - again not an exact degree;
14 \u003d 7 · 2 - again not an exact degree;

Note also that the primes themselves are always exact powers of themselves.

Decimal logarithm

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

The decimal logarithm of x is the logarithm base 10, i.e. the power to which the number 10 must be raised 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 a phrase like “Find lg 0.01” appears in a textbook, you should know: 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 \u003d 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 way, it is even more important than decimal. This is the natural logarithm.

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

Many will ask: what else is the number e? This is an irrational number, its exact meaning cannot be found and written down. I will give only the first figures:
e \u003d 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 \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. Except, of course, units: ln 1 \u003d 0.

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

(from the Greek λόγος - "word", "relation" and ἀριθμός - "number") numbers b by reason a (log α b) is called such a number cand b= a c, that is, log α b=c and b \u003d a c are equivalent. The logarithm makes sense if a\u003e 0, and ≠ 1, b\u003e 0.

In other words logarithm numbers b by reason andis formulated as an indicator of the degree to which the number must be raised ato get the number b(only positive numbers have a logarithm).

This formulation implies that the computation x \u003d log α b, is equivalent to solving the equation a x \u003d b.

For example:

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

We emphasize that the indicated formulation of the logarithm makes it possible to immediately determine logarithm value, when the number under the sign of the logarithm is some degree of the base. And in truth, the formulation of the logarithm makes it possible to prove that if b \u003d a c, then the logarithm of the number b by reason a is equal from... It is also clear that the topic of logarithm is closely related to the topic degree of number.

Calculation of the logarithm is called by taking the logarithm... Taking the logarithm is the mathematical operation of taking the logarithm. When taking the logarithm, the product of the factors is transformed into the sum of the terms.

Potentiation is a mathematical operation inverse to logarithm. In potentiation, the given base is raised to the power of the expression over which the potentiation is performed. In this case, the sums of the members are transformed into the product of the factors.

Real logarithms with bases 2 (binary), e Euler's number e ≈ 2.718 (natural logarithm) and 10 (decimal) are often used.

At this stage, it is advisable to consider samples of logarithmslog 7 2 , ln √ 5, lg0.0001.

And the entries lg (-3), log -3 3.2, log -1 -4.3 do not make sense, since in the first of them a negative number is placed under the sign of the logarithm, in the second - a negative number at the base, and in the third - a negative number under the sign of the logarithm and one at the base.

Conditions for determining the logarithm.

It is worth considering separately the conditions a\u003e 0, a ≠ 1, b\u003e 0 under which definition of the logarithm. Let's consider why these restrictions are taken. An equality of the form x \u003d log α b , called the basic logarithmic identity, which directly follows from the definition of a logarithm given above.

Let's take the condition a ≠ 1... Since one is equal to one to any degree, the equality x \u003d log α b can exist only when b \u003d 1but log 1 1 will be any real number. To eliminate this ambiguity, we take a ≠ 1.

Let us prove the necessity of the condition a\u003e 0... When a \u003d 0 according to the formulation of the logarithm, it can only exist for b \u003d 0... And accordingly then log 0 0can be any nonzero real number, since zero in any nonzero degree is zero. To exclude this ambiguity is given by the condition a ≠ 0... And when a<0 we would have to reject the analysis of rational and irrational values \u200b\u200bof the logarithm, since a degree with a rational and irrational exponent is defined only for non-negative grounds. It is for this reason that the condition is stipulated a\u003e 0.

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

Features of logarithms.

Logarithms characterized by distinctive features, which led to their widespread use to significantly facilitate painstaking calculations. In the transition "to the world of logarithms" multiplication is transformed into a much easier addition, division into subtraction, and exponentiation and root extraction are transformed, respectively, into multiplication and division by an exponent.

The formulation of logarithms and a table of their values \u200b\u200b(for trigonometric functions) were first published in 1614 by the Scottish mathematician John Napier. Logarithmic tables, magnified and detailed by other scientists, were widely used in scientific and engineering calculations, and remained relevant until electronic calculators and computers were used.

The basic properties of the logarithm, the graph of the logarithm, the domain of definition, the set of values, the basic formulas, increase and decrease are given. Finding the derivative of the logarithm is considered. As well as integral, power series expansion and representation by means of complex numbers.

Content

Range of definition, many values, increasing, decreasing

The logarithm is a monotonic function, therefore it has no extrema. The main properties of the logarithm are presented in the table.

Domain	0 < x < + ∞	0 < x < + ∞
Range of values	- ∞ < y < + ∞	- ∞ < y < + ∞
Monotone	increases monotonically	decreases monotonically
Zeros, y \u003d 0	x \u003d 1	x \u003d 1
Points of intersection with the y-axis, x \u003d 0	not	not
	+ ∞	- ∞
	- ∞	+ ∞

Private values

Logarithm base 10 is called decimal logarithm and denoted as follows:

Logarithm base e called natural logarithm:

Basic formulas for logarithms

Logarithm properties following from the definition of the inverse function:

The main property of logarithms and its consequences

Base replacement formula

Taking the logarithm is the mathematical operation of taking the logarithm. When taking the logarithm, the products of the factors are converted to the sum of the terms.
Potentiation is the inverse mathematical operation of taking logarithms. In potentiation, the given base is raised to the power of the expression over which the potentiation is performed. In this case, the sums of the members are converted into products of factors.

Proof of the main formulas for logarithms

Formulas related to logarithms follow from formulas for exponential functions and from the definition of an inverse function.

Consider the property of the exponential function
.
Then
.
Let's apply the exponential function property
:
.

Let us prove the base change formula.
;
.
Setting c \u003d b, we have:

Inverse function

The inverse for a logarithm to base a is exponential function with exponent a.

If, then

If, then

Derivative of the logarithm

Derivative of the logarithm of the modulus x:
.
Derivative of the nth order:
.
Derivation of formulas\u003e\u003e\u003e

To find the derivative of the logarithm, it must be reduced to the base e.
;
.

Integral

The integral of the logarithm is calculated by integrating by parts:.
So,

Expressions in terms of complex numbers

Consider the complex number function z:
.
Let us express the complex number z via module r and the argument φ :
.
Then, using the properties of the logarithm, we have:
.
Or

However, the argument φ not uniquely defined. If we put
, where n is an integer,
it will be the same number for different n.

Therefore, the logarithm, as a function of a complex variable, is not an unambiguous function.

Power series expansion

At the decomposition takes place:

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

See also: