What is called the decimal logarithm. Decimal logarithm and its properties

DEFINITION

Decimal logarithm called the logarithm base 10:

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This logarithm is the solution to the exponential equation. Sometimes (especially in foreign literature) the decimal logarithm is also designated as, although the first two designations are inherent in the natural logarithm.

The first tables of decimal logarithms were published by the English mathematician Henry Briggs (1561-1630) in 1617 (therefore, foreign scientists often call decimal logarithms even Briggs), but these tables contained errors. On the basis of the tables (1783) of the Slovenian and Austrian mathematicians Georg Bartalomeus Vega (Yuri Vekha or Vehovec, 1754-1802), in 1857 the German astronomer and surveyor Karl Bremiker (1804-1877) published the first error-free edition. With the participation of the Russian mathematician and teacher Leonty Filippovich Magnitsky (Telyatin or Telyashin, 1669-1739), the first tables of logarithms were published in Russia in 1703. Decimal logarithms were widely used for calculations.

Decimal Logarithm Properties

This logarithm has all the properties of an arbitrary base logarithm:

1. Basic logarithmic identity:

5. .

7. Transition to a new foundation:

The decimal logarithm function is a function. The plot of this curve is often called logarithmic.

Properties of the function y \u003d lg x

1) Scope of definition:.

2) Lots of values:.

3) General function.

4) The function is non-periodic.

5) The graph of the function intersects the abscissa at a point.

6) Intervals of constancy: title \u003d "(! LANG: Rendered by QuickLaTeX.com" height="16" width="44" style="vertical-align: -4px;"> для !} thats for.

Range of acceptable values \u200b\u200b(ODV) of the logarithm

Now let's talk about constraints (ODZ is the range of allowed values \u200b\u200bof variables).

We remember that, for example, the square root cannot be extracted from negative numbers; or if we have a fraction, then the denominator cannot be zero. Logarithms have similar restrictions:

That is, both the argument and the base must be greater than zero, and the base also cannot be equal.

Why is that?

Let's start with a simple one: let's say that. Then, for example, the number does not exist, since no matter what degree we raise, it always turns out. Moreover, it does not exist for any. But at the same time it can be equal to anything (for the same reason - to any extent equal). Therefore, the object is not of any interest, and it was simply thrown out of mathematics.

We have a similar problem in the case: in any positive degree it is, but it cannot be raised to a negative degree at all, since division by zero will turn out (remember 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 does not exist.

Therefore, it is easier to throw away negative grounds than to mess with them.

Well, since the base a we have only positive, then no matter what degree we raise it, we will always get a strictly positive number. Hence, the argument must be positive. For example, it does not exist, since it will not be a negative number in any way (and even zero, therefore it does not exist either).

In problems with logarithms, the first step is to write down the ODV. Here's an example:

Let's solve the equation.

Remember the definition: a logarithm is the degree to which the base must be raised to get an argument. And by 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, these are numbers and.

But if you immediately take and write down 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 "outside".

To avoid such unpleasant tricks, you need to write down the ODV before you start solving 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.

Decision:

First of all, let's write the ODZ:

Now let's remember what a logarithm is: to what degree should the base be raised to get an argument? Second. I.e:

It would seem that the smaller root is equal. But this is not so: according to the ODZ, the root is external, that is, it is not at all the root of the given equation. Thus, the equation has only one root:.

Answer: .

Basic logarithmic identity

Let's remember the definition of the logarithm in general:

Let's substitute in the second equality instead of the logarithm:

This equality is called the basic logarithmic identity... Although in essence this equality is simply written differently definition of logarithm:

This is the degree to which you have to raise in order to receive.

For instance:

Solve the following examples:

Example 2.

Find the meaning of the expression.

Decision:

Let's recall the rule from the section: that is, when raising a power to a power, the indicators are multiplied. Let's apply it:

Example 3.

Prove that.

Decision:

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 it will be possible to calculate the value. The easiest way to do this is knowing 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:

Evidence:

Let, then.

We have:, h.t.d.

Property 2: Sum of logarithms

The sum of the logarithms with the same bases is equal to the logarithm of the product: .

Evidence:

Let, then. Let, then.

Example:Find the meaning of the expression:.

Decision: .

The formula you just learned helps to simplify the sum of the logarithms, not the difference, so these logarithms cannot be combined immediately. But you can do the opposite - “split” the first logarithm into two: And here is the promised simplification:
.
Why is this needed? Well, for example: what does it matter?

It is now obvious that.

Now simplify yourself:

Tasks:

Answers:

Property 3: Difference of logarithms:

Evidence:

Everything is exactly the same as in point 2:

Let, then.

Let, then. We have:

The example from the last paragraph now becomes even simpler:

A more complicated example:. Can you guess how to decide?

It should be noted here that we do not have a single formula about the logarithms squared. This is something akin to an expression - this cannot be simplified right away.

Therefore, we will digress from the formulas about logarithms, and think about what formulas we use in mathematics most often? Even starting from the 7th grade!

It - . You need to get used to the fact that they are everywhere! And in exponential, and in trigonometric, and in irrational problems, they are found. Therefore, they must be remembered.

If you look closely at the first two terms, it becomes clear that this difference of squares:

Answer for verification:

Simplify yourself.

Examples of

Answers.

Property 4: Removing the exponent from the logarithm argument:

Evidence:And here we also use the definition of a logarithm: let, then. We have:, h.t.d.

You can understand this rule like this:

That is, the degree of the argument is put ahead of the logarithm, as a coefficient.

Example:Find the meaning of the expression.

Decision: .

Decide for yourself:

Examples:

Answers:

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

Evidence:Let, then.

We have:, h.t.d.
Remember: from grounds the degree is rendered as reverse number, unlike the previous case!

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

Or if the degrees are the same:.

Property 7: Transition to a new base:

Evidence:Let, then.

We have:, h.t.d.

Property 8: Swap base and logarithm argument:

Evidence:This is a special case of formula 7: if we substitute, we get:, p.t.d.

Let's look at a few more examples.

Example 4.

Find the meaning of the expression.

We use property of logarithms number 2 - the sum of logarithms with the same base is equal to the logarithm of the product:

Example 5.

Find the meaning of the expression.

Decision:

We use the property of logarithms # 3 and # 4:

Example 6.

Find the meaning of the expression.

Decision:

Using property # 7 - move on to base 2:

Example 7.

Find the meaning of the expression.

Decision:

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On the exam and exam and in general in life

\\ (a ^ (b) \u003d c \\) \\ (\\ Leftrightarrow \\) \\ (\\ log_ (a) (c) \u003d b \\)

Let's explain it easier. For example, \\ (\\ log_ (2) (8) \\) is equal to the degree to which \\ (2 \\) must be raised to get \\ (8 \\). Hence it is clear that \\ (\\ log_ (2) (8) \u003d 3 \\).

Examples:		\\ (\\ log_ (5) (25) \u003d 2 \\)		since \\ (5 ^ (2) \u003d 25 \\)
		\\ (\\ log_ (3) (81) \u003d 4 \\)		since \\ (3 ^ (4) \u003d 81 \\)
		\\ (\\ log_ (2) \\) \\ (\\ frac (1) (32) \\) \\ (\u003d - 5 \\)		since \\ (2 ^ (- 5) \u003d \\) \\ (\\ frac (1) (32) \\)

Logarithm argument and base

Any logarithm has the following "anatomy":

The argument of the logarithm is usually written at its level, and the base is in subscript closer to the sign of the logarithm. And this entry reads like this: "logarithm of twenty-five to base five."

How do I calculate the logarithm?

To calculate the logarithm, you need to answer the question: to what power should the base be raised to get the argument?

for instance, calculate the logarithm: a) \\ (\\ log_ (4) (16) \\) b) \\ (\\ log_ (3) \\) \\ (\\ frac (1) (3) \\) c) \\ (\\ log _ (\\ sqrt (5)) (1) \\) d) \\ (\\ log _ (\\ sqrt (7)) (\\ sqrt (7)) \\) d) \\ (\\ log_ (3) (\\ sqrt (3)) \\)

a) To what degree should \\ (4 \\) be raised to get \\ (16 \\)? Obviously in the second. Therefore:

\\ (\\ log_ (4) (16) \u003d 2 \\)

\\ (\\ log_ (3) \\) \\ (\\ frac (1) (3) \\) \\ (\u003d - 1 \\)

c) To what degree should \\ (\\ sqrt (5) \\) be raised to get \\ (1 \\)? And what degree makes any number one? Zero, of course!

\\ (\\ log _ (\\ sqrt (5)) (1) \u003d 0 \\)

d) To what degree should \\ (\\ sqrt (7) \\) be raised to get \\ (\\ sqrt (7) \\)? First - any number in the first degree is equal to itself.

\\ (\\ log _ (\\ sqrt (7)) (\\ sqrt (7)) \u003d 1 \\)

e) To what degree should \\ (3 \\) be raised to get \\ (\\ sqrt (3) \\)? From we know that it is a fractional degree, and therefore the square root is the degree \\ (\\ frac (1) (2) \\).

\\ (\\ log_ (3) (\\ sqrt (3)) \u003d \\) \\ (\\ frac (1) (2) \\)

Example : Calculate logarithm \\ (\\ log_ (4 \\ sqrt (2)) (8) \\)

Decision :

\\ (\\ log_ (4 \\ sqrt (2)) (8) \u003d x \\)		We need to find the value of the logarithm, let's designate it as x. Now let's use the definition of a logarithm: \\ (\\ log_ (a) (c) \u003d b \\) \\ (\\ Leftrightarrow \\) \\ (a ^ (b) \u003d c \\)
\\ ((4 \\ sqrt (2)) ^ (x) \u003d 8 \\)		What is the link between \\ (4 \\ sqrt (2) \\) and \\ (8 \\)? Two, because both numbers can be represented by two: \\ (4 \u003d 2 ^ (2) \\) \\ (\\ sqrt (2) \u003d 2 ^ (\\ frac (1) (2)) \\) \\ (8 \u003d 2 ^ (3) \\)
\\ (((2 ^ (2) \\ cdot2 ^ (\\ frac (1) (2)))) ^ (x) \u003d 2 ^ (3) \\)		On the left, we use the properties of the degree: \\ (a ^ (m) \\ cdot a ^ (n) \u003d a ^ (m + n) \\) and \\ ((a ^ (m)) ^ (n) \u003d a ^ (m \\ cdot n) \\)
\\ (2 ^ (\\ frac (5) (2) x) \u003d 2 ^ (3) \\)		The grounds are equal, we pass to the equality of indicators
\\ (\\ frac (5x) (2) \\) \\ (\u003d 3 \\)		Multiply both sides of the equation by \\ (\\ frac (2) (5) \\)
		The resulting root is the value of the logarithm

Answer : \\ (\\ log_ (4 \\ sqrt (2)) (8) \u003d 1,2 \\)

Why did you come up with a logarithm?

To understand this, let's solve the equation: \\ (3 ^ (x) \u003d 9 \\). Just match \\ (x \\) for equality to work. Of course, \\ (x \u003d 2 \\).

Now solve the equation: \\ (3 ^ (x) \u003d 8 \\). What is x? That's just the point.

The most quick-witted will say: "X is slightly less than two." How exactly do you write this number? To answer this question, they came up with a logarithm. Thanks to him, the answer here can be written as \\ (x \u003d \\ log_ (3) (8) \\).

I want to emphasize that \\ (\\ log_ (3) (8) \\), like any logarithm is just a number... Yes, it looks unusual, but short. Because if we wanted to write it as a decimal fraction, then it would look like this: \\ (1.892789260714 ..... \\)

Example : Solve the equation \\ (4 ^ (5x-4) \u003d 10 \\)

Decision :

\\ (4 ^ (5x-4) \u003d 10 \\)		\\ (4 ^ (5x-4) \\) and \\ (10 \u200b\u200b\\) cannot be reduced to the same reason. This means that you cannot do without the logarithm. Let's use the definition of a logarithm: \\ (a ^ (b) \u003d c \\) \\ (\\ Leftrightarrow \\) \\ (\\ log_ (a) (c) \u003d b \\)
\\ (\\ log_ (4) (10) \u003d 5x-4 \\)		Mirror the equation so that x is on the left
\\ (5x-4 \u003d \\ log_ (4) (10) \\)		Before us. Move \\ (4 \\) to the right. And don't be intimidated by the logarithm, treat it like an ordinary number.
\\ (5x \u003d \\ log_ (4) (10) +4 \\)		Divide the equation by 5
\\ (x \u003d \\) \\ (\\ frac (\\ log_ (4) (10) +4) (5) \\)		Here is our root. Yes, it looks strange, but the answer is not chosen.

Answer : \\ (\\ frac (\\ log_ (4) (10) +4) (5) \\)

Decimal and natural logarithms

As stated in the definition of a logarithm, its base can be any positive number other than one \\ ((a\u003e 0, a \\ neq1) \\). And among all the possible reasons, there are two that occur so often that a special short notation has been invented for logarithms with them:

Natural logarithm: a logarithm whose base is Euler's number \\ (e \\) (equal to approximately \\ (2.7182818 ... \\)), and is written as \\ (\\ ln (a) \\)

I.e, \\ (\\ ln (a) \\) is the same as \\ (\\ log_ (e) (a) \\)

Decimal logarithm: A logarithm with base 10 is written \\ (\\ lg (a) \\).

I.e, \\ (\\ lg (a) \\) is the same as \\ (\\ log_ (10) (a) \\), where \\ (a \\) is some number.

Basic logarithmic identity

Logarithms have many properties. One of them is called "Basic Logarithmic Identity" and looks like this:

\\ (a ^ (\\ log_ (a) (c)) \u003d c \\)

This property follows directly from the definition. Let's see how exactly this formula appeared.

Let's remember a short notation of the definition of a logarithm:

if \\ (a ^ (b) \u003d c \\) then \\ (\\ log_ (a) (c) \u003d b \\)

That is, \\ (b \\) is the same as \\ (\\ log_ (a) (c) \\). Then we can write \\ (\\ log_ (a) (c) \\) instead of \\ (b \\) in the formula \\ (a ^ (b) \u003d c \\). It turned out \\ (a ^ (\\ log_ (a) (c)) \u003d c \\) - the main logarithmic identity.

You can find the rest of the properties of logarithms. With their help, you can simplify and calculate the values \u200b\u200bof expressions with logarithms, which are difficult to calculate "head-on".

Example : Find the value of the expression \\ (36 ^ (\\ log_ (6) (5)) \\)

Decision :

Answer : \(25\)

How can a number be written as a logarithm?

As mentioned above, any logarithm is just a number. The converse is also true: any number can be written as a logarithm. For example, we know that \\ (\\ log_ (2) (4) \\) is two. Then you can write \\ (\\ log_ (2) (4) \\) instead of two.

But \\ (\\ log_ (3) (9) \\) is also \\ (2 \\), so you can also write \\ (2 \u003d \\ log_ (3) (9) \\). Similarly, with \\ (\\ log_ (5) (25) \\), and \\ (\\ log_ (9) (81) \\), etc. That is, it turns out

\\ (2 \u003d \\ log_ (2) (4) \u003d \\ log_ (3) (9) \u003d \\ log_ (4) (16) \u003d \\ log_ (5) (25) \u003d \\ log_ (6) (36) \u003d \\ Thus, if we need, we can anywhere (even in an equation, even in an expression, even in an inequality) write two as a logarithm with any base - we just write the base squared as an argument.

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Likewise with a triple - it can be written as \\ (\\ log_ (2) (8) \\), or as \\ (\\ log_ (3) (27) \\), or as \\ (\\ log_ (4) (64) \\) ... Here we write the base in a cube as an argument:

\\ (3 \u003d \\ log_ (2) (8) \u003d \\ log_ (3) (27) \u003d \\ log_ (4) (64) \u003d \\ log_ (5) (125) \u003d \\ log_ (6) (216) \u003d \\ And with a four:

\\ (4 \u003d \\ log_ (2) (16) \u003d \\ log_ (3) (81) \u003d \\ log_ (4) (256) \u003d \\ log_ (5) (625) \u003d \\ log_ (6) (1296) \u003d \\ And with minus one:

\\ (- 1 \u003d \\) \\ (\\ log_ (2) \\) \\ (\\ frac (1) (2) \\) \\ (\u003d \\) \\ (\\ log_ (3) \\) \\ (\\ frac (1) ( 3) \\) \\ (\u003d \\) \\ (\\ log_ (4) \\) \\ (\\ frac (1) (4) \\) \\ (\u003d \\) \\ (\\ log_ (5) \\) \\ (\\ frac (1 ) (5) \\) \\ (\u003d \\) \\ (\\ log_ (6) \\) \\ (\\ frac (1) (6) \\) \\ (\u003d \\) \\ (\\ log_ (7) \\) \\ (\\ frac (1) (7) \\) \\ (... \\)

And with one third:

\\ (\\ frac (1) (3) \\) \\ (\u003d \\ log_ (2) (\\ sqrt (2)) \u003d \\ log_ (3) (\\ sqrt (3)) \u003d \\ log_ (4) (\\ sqrt ( 4)) \u003d \\ log_ (5) (\\ sqrt (5)) \u003d \\ log_ (6) (\\ sqrt (6)) \u003d \\ log_ (7) (\\ sqrt (7)) ... \\)

Any number \\ (a \\) can be represented as a logarithm with base \\ (b \\): \\ (a \u003d \\ log_ (b) (b ^ (a)) \\)

: Find the meaning of the expression

\\ (\\ frac (\\ log_ (2) (14)) (1+ \\ log_ (2) (7)) \\)

Example It is known from the high school curriculum that any positive number can be represented as the number 10 to some extent.

Decision :

Answer : \(1\)

However, this is simply the case when the number is a multiple of 10.

number

100 is 10x10 or 102
Example :

• number 1000 is 10x10x10 or 103and
• etc.
• what if, for example, you need to express the number 8299 as the number 10 to some extent? How to find this number with a certain degree of accuracy, which in this case is 3.919 ...?output is logarithm and logarithmic tables

Knowledge of logarithms and the ability to use logarithmic tables can greatly simplify many complex arithmetic operations. Decimal logarithms are convenient for practical use.

Historical reference

The principle underlying any system of logarithms has been known for a very long time and can be traced back to the depths of history up to ancient Babylonian mathematics (about 2000 BC). However, the first tables of logarithms were compiled independently of each other by the Scottish mathematician HUJ. Napier (1550-1617) U H and the Swiss I. Burgi (1552-1632). The first tables of decimal logarithms were compiled and published by the English mathematician H. Briggs (1561-1630).

We suggest the reader, without going deeply into the mathematical essence of the issue, to remember or restore in memory several simple definitions, conclusions and formulas:.
Definition of logarithm

and.

• The logarithm of a given number is the exponent to which another number must be raised, called the base of the logarithm (and

) to get the given number.for any reason, the logarithm of one is zero: a0 \u003d 1

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• Negative numbers don't have logarithms
• Every positive number has a logarithm
• For radix greater than 1, logarithms of numbers less than 1 are negative, and logarithms of numbers greater than 1 are positive
• Logarithm of base is 1
• Larger number corresponds to larger logarithm
• As the number increases from 0 to 1, its logarithm increases from-∞ to 0; as the number increases from 1 to+∞ its logarithm increases from 1 to+∞ (where, ±∞ - a sign adopted in mathematics to denote negative or positive infinity of numbers)
• For practical use, logarithms are convenient, the base of which is the number 10

These logarithms are called decimal and are denotedlg ... For instance:

• logarithm base 10 of 10 is 1. In other words, the number 10 must be raised to the first power to get the number 10 (101 \u003d 10), that is,lg10 \u003d 1
• logarithm base 10 of 100 is 2. In other words, 10 must be squared to get 100 (102 \u003d 100), that is. lg100 \u003d 2

U Conclusion number 1 U : the logarithm of an integer represented by one followed by zeros, is a positive integer containing as many ones as there are zeros in the image of the number

• logarithm base 10 of 0.1 is -1. In other words, the number 10 must be raised to minus the first power to get the number 0.1 (10-1 \u003d 0.1), i.e.lg0,1 \u003d -1
• logarithm base 10 of 0.01 is -2. In other words, the number 10 must be raised to the minus second power to get the number 0.1 (10-2 \u003d 0.01), i.e.lg0.01 \u003d -2

U Conclusion number 2 U : the logarithm of a decimal fraction, represented by one with leading zeros, is an integer negative number containing as many negative ones as there are zeros in the image of the fraction, counting, including 0, integers

• in accordance with definition # 1 (see above):

lg1 \u003d 0

• the logarithm of 8300 to base 10 is 3.9191 ... In other words, the number 10 needs to be raised to the power of 3.9191 ... to get the number 8300 (103.9191 ... \u003d 8300), i.e. lg8300 \u003d 3.9191 ...

U Conclusion number 3 U : the logarithm of a number that is not expressed by one followed by zeros is an irrational number and, therefore, cannot be expressed exactly by means of numbers.
Usually, irrational logarithms are expressed approximately in the form of a decimal fraction with several decimal places. The whole number of this fraction (even if it was "0 integers") is called characteristic, and the fractional part is mantissa logarithm. If, for example, the logarithm is 1,5441 , then its characteristic is 1 , and the mantissa is 0,5441 .

• Basic properties of logarithms, incl. decimal:
• the logarithm of the product is equal to the sum of the logarithms of the factors:lg ( a. b) \u003d lga + lgb
• the logarithm of the quotient is equal to the logarithm of the dividend without the logarithm of the divisor, i.e. the logarithm of the fraction is equal to the logarithm of the numerator without the logarithm of the denominator:

• logarithms of two reciprocal numbers in the same base differ from each other only in sign

• the logarithm of the power is equal to the product of the exponent by the logarithm of its base, i.e. the logarithm of a power is equal to the exponent of this power multiplied by the logarithm of the number raised to the power:

lg ( bk) \u003d k. lg b

To finally understand what the decimal logarithm of an arbitrary number is, consider in detail several examples.

U Example No. 2.1.1 U.
Take an integer like 623 and a mixed number like 623.57.
We know that the logarithm of a number consists of a characteristic and a mantissa.
Let's count how many digits are in a given whole number, or in the whole part of a mixed number. In our examples, these numbers are 3.
Therefore, each of the numbers 623 and 623.57 is greater than 100, but less than 1000.
Thus, we can conclude that the logarithm of each of these numbers will be greater than lg 100, that is, more than 2, but less than lg 1000, that is, less than 3 (remember that a larger number has a larger logarithm).
Hence:
lg 623 \u003d 2, ...
lg 623.57 \u003d 2, ...
(dots replace unknown mantissas).

U Conclusion number 4 U : decimal logarithms have the convenience that their characteristics can always be found by one type of number .

Suppose, in general, in a given integer, or in an integer part of a given mixed number, there are m digits. Since the smallest integer containing m digits is one with m-1 zeros at the end, then (denoting this number N) we can write the inequality:

hence,
m-1< lg N < m,
so
lg N \u003d (m-1) + positive fraction.
means
characteristic lgN \u003d m-1

U Conclusion number 5 U : the characteristic of the decimal logarithm of an integer or mixed number contains as many positive ones as there are digits in the whole part of the number without one.

U Example No. 2.1.2.

Now let's take a few decimal fractions, i.e. numbers less than 1 (in other words, having 0 integers):
0.35; 0.07; 0.0056; 0.0008, etc.
The logarithms of each of these numbers will be between two negative integers that differ by one unit. Moreover, each of them is equal to the smaller of these negative numbers, increased by some positive fraction.
For instance,
lg0.0056 \u003d -3 + positive fraction
In this case, the positive fraction will be 0.7482.
Then:
lg 0.0056 \u003d -3 + 0.7482
U Notes U:
Sums such as -3 + 0.7482, consisting of a negative integer and a positive decimal fraction, agreed to write in logarithmic calculations in abbreviated form as follows:
,7482
(such a number is read: with a minus, 7482 ten-thousandths), that is, they put a minus sign above the characteristic in order to show that it refers only to this characteristic, and not to the mantissa, which remains positive.

Thus, the above numbers can be written as decimal logarithms
lg 0.35 \u003d, ...
lg 0.07 \u003d, ...
lg 0.00008 \u003d, ...
Let, in general, the number A be a decimal fraction, which has m zeros before the first significant digit α, including 0 integers:

then it is obvious that

Hence:

i.e.
-m< log A < -(m-1).
Since from two integers:
-m and - (m-1) smaller is -m
then
lg А \u003d -m + positive fraction

U Conclusion number 6 U : characteristic of the logarithm of a decimal fraction, i.e. numbers less than 1, contains as many negative ones as there are zeros in the decimal fraction before the first significant digit, including zero integers; the mantissa of such a logarithm is positive

Example No. 2.1.3.

Let's multiply some number N (whole or fractional - everything is equal) by 10, by 100 by 1000 ..., generally by 1 with zeros, and see how lg N will change from this.
Since the logarithm of the product is equal to the sum of the logarithms of the factors, then
lg (N.10) \u003d lg N + lg 10 \u003d lg N + 1;
lg (N.100) \u003d lg N + lg 100 \u003d lg N + 2;
lg (N. 1000) \u003d lg N + lg 1000 \u003d lg N + 3, etc.

When we add an integer to lg N, this number is always added to the characteristic; the mantissa always remains unchanged in these cases.

Example
if lg N \u003d 2.7804, then 2.7804 + 1 \u003d 3.7804; 2.7804 + 2 \u003d 4.7801, etc .;
or if lg N \u003d 3.5649, then 3.5649 + 1 \u003d 2.5649; 3.5649 - 2 \u003d 1.5649, etc.

Conclusion number 7 : from multiplying a number by 10, 100, 1000, .., generally by 1 with zeros, the mantissa of the logarithm does not change, and the characteristic increases by as many units as there are zeros in the factor.

Similarly, taking into account that the logarithm of the quotient is equal to the logarithm of the dividend without the logarithm of the divisor, we get:
log N / 10 \u003d log N - log 10 \u003d log N - 1;
log N / 100 \u003d log N - log 100 \u003d log N - 2;
log N / 1000 \u003d log N - log 1000 \u003d log N - 3, etc.
When an integer is subtracted from lg N from the logarithm, subtract this integer always follows from the characteristic, and leave the mantissa unchanged. then we can say:

Conclusion number 8 : From dividing a number by 1 with zeros, the mantissa of the logarithm does not change, and the characteristic decreases by as many units as there are zeros in the divisor.

Conclusion number 9 : The mantissa of the logarithm of a decimal number does not change from carrying a comma, because carrying a comma is equivalent to multiplying or dividing by 10, 100, 1000, etc.

Thus, the logarithms of numbers are:
0,00423, 0,0423, 4,23, 423
differ only in characteristics, but not in mantissas (provided that all mantissas are positive).

Conclusion number 9 : the mantissa of numbers that have the same significant part, but differ only in zeros at the end, are the same: for example, the logarithms of numbers: 23, 230, 2300, 23,000 differ only in characteristics.