What is a decimal logarithm? Logarithm - properties, formulas, graph Logarithm 6 to base 10.

22.03.2023 | For children

The power of a given number is a mathematical term coined centuries ago. In geometry and algebra, there are two options - decimal and natural logarithms. They are calculated by different formulas, while equations that differ in spelling are always equal to each other. This identity characterizes the properties that relate to the useful potential of the function.

Features and important signs

There are currently ten known mathematical qualities. The most common and popular of them are:

The radical log divided by the magnitude of the root is always the same as the decimal logarithm √.
The product log is always equal to the producer's sum.
Lg = the magnitude of the power multiplied by the number that is raised to it.
If you subtract the divisor from log of the dividend, you get log of the quotient.

In addition, there is an equation based on the main identity (considered the key), a transition to an updated basis, and several minor formulas.

Calculating the decimal logarithm is a fairly specialized task, so integrating properties into a solution must be approached carefully and regularly checked your actions and consistency. We must not forget about the tables, which must be constantly consulted, and be guided only by the data found there.

Varieties of mathematical term

Main differences mathematical number"hidden" at the base (a). If it has an exponent of 10, then it is log decimal. In the opposite case, “a” is transformed into “y” and has transcendental and irrational characteristics. It is also worth noting that the natural value is calculated by a special equation, where the proof is a theory studied outside school curriculum senior classes.

Decimal logarithms are widely used in the calculation of complex formulas. Entire tables have been compiled to facilitate calculations and clearly show the process of solving the problem. In this case, before going directly to the matter, you need to raise log to In addition, in each store school supplies You can find a special ruler with a printed scale that helps you solve an equation of any complexity.

The decimal logarithm of a number is called Brigg's number, or Euler's number, in honor of the researcher who first published the quantity and discovered the contrast between the two definitions.

Two types of formula

All types and varieties of problems for calculating the answer, having the term log in the condition, have a separate name and a strict mathematical structure. Exponential equation is an almost exact copy of logarithmic calculations, if you look at the correctness of the solution. It’s just that the first option includes a specialized number that helps you quickly understand the condition, and the second replaces log with an ordinary power. In this case, calculations using the last formula must include a variable value.

Difference and terminology

Both main indicators have their own characteristics that distinguish the numbers from each other:

Decimal logarithm. An important detail of the number is the mandatory presence of a base. The standard version of the value is 10. It is marked with the sequence - log x or log x.
Natural. If its base is the sign "e", which is a constant identical to a strictly calculated equation, where n rapidly moves towards infinity, then the approximate size of the number in digital equivalent is 2.72. The official marking, adopted both in school and in more complex professional formulas, is ln x.
Different. In addition to basic logarithms, there are hexadecimal and binary types (base 16 and 2, respectively). There is an even more complex option with a base indicator of 64, which falls under systematic adaptive type control, with geometric accuracy calculating the final result.

The terminology includes the following quantities included in the algebraic problem:

meaning;
argument;
base.

Calculating log number

There are three ways to quickly and verbally make all the necessary calculations to find the result of interest, with the obligatory correct outcome of the solution. Initially, we bring the decimal logarithm closer to its order (the scientific notation of a number to a power). Each positive value can be specified by an equation, where it is equal to the mantissa (a number from 1 to 9) multiplied by ten in nth degree. This calculation option is based on two mathematical facts:

the product and sum log always have the same exponent;
the logarithm taken from a number from one to ten cannot exceed a value of 1 point.

If an error in the calculation does occur, then it is never less than one in the direction of subtraction.
Accuracy increases if you consider that lg with base three has a final result of five tenths of one. Therefore, any mathematical value greater than 3 automatically adds one point to the answer.
Almost perfect accuracy is achieved if you have a specialized table at hand that can be easily used in your assessment activities. With its help you can find out what the decimal logarithm is equal to tenths of a percent of the original number.

History of real log

The sixteenth century was in dire need of more complex calculus than was known to science at the time. This was especially true for dividing and multiplying multi-digit numbers with great consistency, including fractions.

At the end of the second half of the era, several minds immediately came to the conclusion about adding numbers using a table that compared two and a geometric one. In this case, all basic calculations had to rest on the last value. Scientists have integrated subtraction in the same way.

The first mention of lg took place in 1614. This was done by an amateur mathematician named Napier. It is worth noting that, despite the enormous popularization of the results obtained, an error was made in the formula due to ignorance of some definitions that appeared later. It began with the sixth digit of the indicator. The closest to the understanding of the logarithm were the Bernoulli brothers, and the debut legalization occurred in the eighteenth century by Euler. He also extended the function to the field of education.

History of complex log

Debut attempts to integrate lg into the general public were made at the dawn of the 18th century by Bernoulli and Leibniz. But they were never able to draw up comprehensive theoretical calculations. There was a whole discussion about this, but precise definition the number was not assigned. Later the dialogue resumed, but between Euler and d'Alembert.

The latter agreed in principle with many of the facts proposed by the founder of the value, but believed that positive and negative indicators should be equal. In the middle of the century the formula was demonstrated as a final version. In addition, Euler published the derivative of the decimal logarithm and compiled the first graphs.

Tables

The properties of numbers indicate that multi-digit numbers can not be multiplied, but their log can be found and added using specialized tables.

This indicator has become especially valuable for astronomers who are forced to work with a large set of sequences. IN Soviet time The decimal logarithm was looked for in Bradis's collection, published in 1921. Later, in 1971, the Vega edition appeared.

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 a degree by a rational exponent, to which the definition of a logarithm is reduced.
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 considering only numerical expressions, where it is not required to know the VA of the logarithm. 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 factor it 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 is a 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. It's about 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 will not 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 is irrational. Except, of course, for one: ln 1 = 0.

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

The basic properties of the logarithm, logarithm graph, domain of definition, set of values, basic formulas, increasing and decreasing are given. Finding the derivative of a logarithm is considered. As well as integral, power series expansion and representation using complex numbers.

Content

Domain, set of values, increasing, decreasing

The logarithm is a monotonic function, so 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	monotonically increases	monotonically decreases
Zeros, y = 0	x = 1	x = 1
Intercept points with the ordinate axis, x = 0	No	No
	+ ∞	- ∞
	- ∞	+ ∞

Private values

The logarithm to base 10 is called decimal logarithm and is denoted as follows:

Logarithm to base e called natural logarithm :

Basic formulas for logarithms

Properties of the logarithm arising from the definition of the inverse function:

The main property of logarithms and its consequences

Base replacement formula

Logarithm is the mathematical operation of taking a logarithm. When taking logarithms, products of factors are converted into sums of terms.
Potentiation is the mathematical operation inverse to logarithm. During potentiation, a given base is raised to the degree of expression over which potentiation is performed. In this case, the sums of terms are transformed into products of factors.

Proof of basic 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 property of the exponential function
:
.

Let us prove the base replacement formula.
;
.
Assuming c = b, we have:

Inverse function

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

If , then

Derivative of logarithm

Derivative of the logarithm of modulus x:
.
Derivative of nth order:
.
Deriving formulas > > >

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

Integral

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

Expressions using complex numbers

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

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

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

Power series expansion

When the expansion takes place:

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.