Decimal logarithm 1 2. Logarithm

Instructions

Write down the given logarithmic expression. If the expression uses the logarithm of 10, then its notation is shortened and looks like this: lg b is the decimal logarithm. If the logarithm has the number e as its base, then write the expression: ln b – natural logarithm. It is understood that the result of any is the power to which the base number must be raised to obtain the number b.

When finding the sum of two functions, you simply need to differentiate them one by one and add the results: (u+v)" = u"+v";

When finding the derivative of the product of two functions, it is necessary to multiply the derivative of the first function by the second and add the derivative of the second function multiplied by the first function: (u*v)" = u"*v+v"*u;

In order to find the derivative of the quotient of two functions, it is necessary to subtract from the product of the derivative of the dividend multiplied by the divisor function the product of the derivative of the divisor multiplied by the function of the dividend, and divide all this by the divisor function squared. (u/v)" = (u"*v-v"*u)/v^2;

If given complex function, then it is necessary to multiply the derivative of internal function and the derivative of the external one. Let y=u(v(x)), then y"(x)=y"(u)*v"(x).

Using the results obtained above, you can differentiate almost any function. So let's look at a few examples:

y=x^4, y"=4*x^(4-1)=4*x^3;

y=2*x^3*(e^x-x^2+6), y"=2*(3*x^2*(e^x-x^2+6)+x^3*(e^x-2 *x));
There are also problems involving calculating the derivative at a point. Let the function y=e^(x^2+6x+5) be given, you need to find the value of the function at the point x=1.
1) Find the derivative of the function: y"=e^(x^2-6x+5)*(2*x +6).

2) Calculate the value of the function in given point y"(1)=8*e^0=8

Video on the topic

Useful advice

Learn the table of elementary derivatives. This will significantly save time.

Sources:

• derivative of a constant

So, what is the difference between an irrational equation and a rational one? If the unknown variable is under the sign square root, then the equation is considered irrational.

Instructions

The main method for solving such equations is the method of constructing both sides equations into a square. However. this is natural, the first thing you need to do is get rid of the sign. This method is not technically difficult, but sometimes it can lead to trouble. For example, the equation is v(2x-5)=v(4x-7). By squaring both sides you get 2x-5=4x-7. Solving such an equation is not difficult; x=1. But the number 1 will not be given equations. Why? Substitute one into the equation instead of the value of x. And the right and left sides will contain expressions that do not make sense, that is. This value is not valid for a square root. Therefore, 1 is an extraneous root, and therefore this equation has no roots.

So, an irrational equation is solved using the method of squaring both its sides. And having solved the equation, it is necessary to cut off extraneous roots. To do this, substitute the found roots into the original equation.

Consider another one.
2х+vх-3=0
Of course, this equation can be solved using the same equation as the previous one. Move Compounds equations, which do not have a square root, to the right side and then use the squaring method. solve the resulting rational equation and roots. But also another, more elegant one. Enter a new variable; vх=y. Accordingly, you will receive an equation of the form 2y2+y-3=0. That is, the usual quadratic equation. Find its roots; y1=1 and y2=-3/2. Next, solve two equations vх=1; vх=-3/2. The second equation has no roots; from the first we find that x=1. Don't forget to check the roots.

Solving identities is quite simple. To do this, it is necessary to carry out identical transformations until the set goal is achieved. Thus, with the help of the simplest arithmetic operations the task at hand will be solved.

You will need

• - paper;
• - pen.

Instructions

The simplest of such transformations are algebraic abbreviated multiplications (such as the square of the sum (difference), difference of squares, sum (difference), cube of the sum (difference)). In addition, there are many and trigonometric formulas, which are essentially the same identities.

Indeed, the square of the sum of two terms is equal to the square of the first plus twice the product of the first by the second and plus the square of the second, that is, (a+b)^2= (a+b)(a+b)=a^2+ab +ba+b ^2=a^2+2ab+b^2.

Simplify both

General principles of the solution

Repeat according to the textbook mathematical analysis or higher mathematics, what a definite integral is. As is known, the solution definite integral there is a function whose derivative gives an integrand. This function is called antiderivative. Based on this principle, the main integrals are constructed.
Determine by the type of the integrand which of the table integrals is suitable in this case. It is not always possible to determine this immediately. Often, the tabular form becomes noticeable only after several transformations to simplify the integrand.

Variable Replacement Method

If the integrand function is trigonometric function, whose argument contains some polynomial, then try using the variable replacement method. In order to do this, replace the polynomial in the argument of the integrand with some new variable. Based on the relationship between the new and old variables, determine the new limits of integration. By differentiating this expression, find the new differential in . So you will get new look of the previous integral, close to or even corresponding to any tabular one.

Solving integrals of the second kind

If the integral is an integral of the second kind, a vector form of the integrand, then you will need to use the rules for the transition from these integrals to scalar ones. One such rule is the Ostrogradsky-Gauss relation. This law allows us to move from the rotor flux of a certain vector function to the triple integral over the divergence of a given vector field.

Substitution of integration limits

After finding the antiderivative, it is necessary to substitute the limits of integration. First, substitute the value of the upper limit into the expression for the antiderivative. You will get some number. Next, subtract from the resulting number another number obtained from the lower limit into the antiderivative. If one of the limits of integration is infinity, then when substituting it into antiderivative function it is necessary to go to the limit and find what the expression strives for.
If the integral is two-dimensional or three-dimensional, then you will have to represent the limits of integration geometrically to understand how to evaluate the integral. Indeed, in the case of, say, a three-dimensional integral, the limits of integration can be entire planes that limit the volume being integrated.

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, and 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

On at the moment distinguish 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

The main differences of the mathematical number are “hidden” in 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 obtained wide application when calculating 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 build 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.

Decimal logarithm The number is called Brigg's number, or Euler's number, in honor of the researcher who first published the value 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 practically an exact copy logarithmic calculations, if viewed from the perspective of 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 is rapidly moving 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 a systematic adaptive type control that calculates the final result with geometric accuracy.

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.

1. If an error in the calculation does occur, then it is never less than one in the direction of subtraction.
2. 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.
3. 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 Bernoulli brothers were closest to understanding the logarithm, and the debut legitimization 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 era The decimal logarithm was looked for in Bradis's collection, published in 1921. Later, in 1971, the Vega edition appeared.

SECTION XIII.

LOGARITHMAS AND THEIR APPLICATIONS.

§ 2. Decimal logarithms.

The decimal logarithm of the number 1 is 0. Decimal logarithms of positive powers of 10, i.e. numbers 10, 100, 1000,.... essentially, positive numbers 1, 2, 3,...., so in general the logarithm of a number denoted by one with zeros, equal to the number zeros. Decimal logarithms of negative powers of 10, i.e. fractions 0.1, 0.01, 0.001,.... are negative numbers -1, -2, -3....., so in general a logarithm decimal with a numerator of one is equal to the negative number of zeros of the denominator.

The logarithms of all other commensurable numbers are incommensurable. Such logarithms are calculated approximately, usually with an accuracy of one hundred thousandth, and therefore are expressed in five-digit decimal fractions; for example, log 3 = 0.47712.

When presenting the theory of decimal logarithms, all numbers are assumed to be composed according to the decimal system of their units and fractions, and all logarithms are expressed through a decimal fraction containing 0 integers, with an integer increase or decrease. The fractional part of the logarithm is called its mantissa, and the whole increase or decrease is called its characteristic. Logarithms of numbers greater than one are always positive and therefore have a positive characteristic; logarithms of numbers less than one are always negative, but they are represented in such a way that their mantissa turns out to be positive, and one characteristic is negative: for example, log 500 = 0.69897 + 2 or shorter 2.69897, and log 0.05 = 0, 69897-2, which for brevity is denoted as 2.69897, putting the characteristic in place of integers, but with a sign above it. Thus, the logarithm of a number greater than one represents the arithmetic sum of a positive integer and a positive fraction, and the logarithm of a number less than one represents the algebraic sum of a negative integer with a positive fraction.

Any negative logarithm can be reduced to the indicated artificial form. For example, we have log 3 / 5 = log 3 - log 5 = 0.47712-0.69897 = -0.22185. To convert this true logarithm into an artificial form, we add 1 to it and, after algebraic addition, we indicate the subtraction of one for correction.

We get log 3 / 5 = log 0.6 = (1-0.22185)-1 = 0.77815-1. It turns out that the mantissa 0.77815 is the same one that corresponds to the numerator 6 of this number, represented in the decimal system in the form of the fraction 0.6.

In the indicated representation of decimal logarithms, their mantissa and characteristics have important properties in connection with the designation of the numbers corresponding to them in the decimal system. To explain these properties, we note the following. Let us take as the main type of number some arbitrary number contained between 1 and 10, and, expressing it in the decimal system, present it in the form a,b,c,d,e,f ...., Where A there is one of significant figures 1, 2, 3, 4, 5, 6, 7, 8, 9, and decimal places, b,c,d,e,f ....... are any numbers, between which there may be zeros. Due to the fact that the taken number is contained between 1 and 10, its logarithm is contained between 0 and 1 and therefore this logarithm consists of one mantissa without characteristic or with characteristic 0. Let us denote this logarithm in the form 0 ,α β γ δ ε ...., Where α, β ,γ ,δ, ε the essence of some numbers. Let us now multiply this number on the one hand by the numbers 10, 100, 1000,.... and on the other hand by the numbers 0.1, 0.01, 0.001,... and apply the theorems on the logarithms of the product and the quotient. Then we get a series of numbers greater than one and a series of numbers less than one with their logarithms:

lg A ,bcde f ....= 0 ,α β γ δ ε ....

lg ab,cde f ....= 1 ,α β γ δ ε ....lg 0,abcde f ....= 1 ,α β γ δ ε ....

lg аbc,de f ....= 2 ,α β γ δ ε ....lg 0.0abcde f ....= 2 ,α β γ δ ε ....

lg аbcd,e f ....= 3 ,α β γ δ ε ....lg 0.00abcde f ....= 3 ,α β γ δ ε ....

When considering these equalities, the following properties of the mantissa and characteristics are revealed:

Mantissa property. The mantissa depends on the location and type of the gapping digits of the number, but does not at all depend on the place of the comma in the designation of this number. Mantissas of logarithms of numbers having a decimal ratio, i.e. those whose multiple ratio is equal to any positive or negative degree ten are the same.

Characteristic property. The characteristic depends on the rank of the highest units or decimal fractions of a number, but does not at all depend on the type of digits in the designation of this number.

If we name the numbers A ,bcde f ...., ab,cde f ...., аbc,de f .... numbers of positive digits - first, second, third, etc., digit of number 0,abcde f .... we will consider zero, and the digits of numbers 0.0abcde f ...., 0.00abcde f ...., 0.000abcde f .... if we express negative numbers minus one, minus two, minus three, etc., then we can say in general that the characteristic of the logarithm of any decimal number per unit less number, indicating the rank

101. Knowing that log 2 =0.30103, find the logarithms of the numbers 20.2000, 0.2 and 0.00002.

101. Knowing that log 3=0.47712, find the logarithms of the numbers 300, 3000, 0.03 and 0.0003.

102. Knowing that log 5 = 0.69897, find the logarithms of the numbers 2.5, 500, 0.25 and 0.005.

102. Knowing that log 7 = 0.84510, find the logarithms of the numbers 0.7, 4.9, 0.049 and 0.0007.

103. Knowing log 3=0.47712 and log 7=0.84510, find the logarithms of the numbers 210, 0.021, 3/7, 7/9 and 3/49.

103. Knowing log 2=0.30103 and log 7=0.84510, find the logarithms of the numbers 140, 0.14, 2/7, 7/8 and 2/49.

104. Knowing log 3 = 0.47712 and log 5 = O.69897, find the logarithms of the numbers 1.5, 3 / 5, 0.12, 5 / 9 and 0.36.

104. Knowing log 5 = 0.69897 and log 7 = 0.84510, find the logarithms of the numbers 3.5, 5 / 7, 0.28, 5 / 49 and 1.96.

Decimal logarithms of numbers expressed in no more than four digits are found directly from the tables, and from the tables the mantissa of the desired logarithm is found, and the characteristic is set in accordance with the rank of the given number.

If the number contains more than four digits, then finding the logarithm is accompanied by an additional calculation. The rule is: to find the logarithm of a number containing more than four digits, you need to find in the tables the number indicated by the first four digits and write the mantissa corresponding to these four digits; then multiply the tabular difference of the mantissa by the number made up of the discarded digits, in the product, discard as many digits from the right as were discarded in the given number, and add the result to the last digits of the found mantissa; put the characteristic in accordance with the rank of the given number.

When a number is searched for using a given logarithm and this logarithm is contained in tables, then the digits of the sought number are found directly from the tables, and the rank of the number is determined in accordance with the characteristics of the given logarithm.

If this logarithm is not contained in the tables, then searching for the number is accompanied by an additional calculation. The rule is: to find the number corresponding to a given logarithm, the mantissa of which is not contained in the tables, you need to find the nearest smaller mantissa and write down the digits of the number corresponding to it; then multiply the difference between the given mantissa and the found one by 10 and divide the product by the tabulated difference; add the resulting digit of the quotient to the right to the written digits of the number, which is why you get the desired set of digits; The rank of the number must be determined in accordance with the characteristics of the given logarithm.

105. Find the logarithms of the numbers 8, 141, 954, 420, 640, 1235, 3907, 3010, 18.43, 2.05, 900.1, 0.73, 0.0028, 0.1008, 0.00005.

105. Find the logarithmic of the numbers 15.154, 837, 510, 5002,1309-, 8900, 8.315, 790.7, 0.09, 0.6745, 0.000745, 0.04257, 0.00071.

106. Find the logarithms of the numbers 2174.6, 1445.7, 2169.5, 8437.2, 46.472, 6.2853, 0.7893B, 0.054294, 631.074, 2.79556, 0.747428, 0.00237158.

106. Find the logarithms of the numbers 2578.4, 1323.6, 8170.5, 6245.3, 437.65, 87.268, 0.059372, 0.84938, 62.5475, 131.037, 0.593946, 0.00234261.

107. Find the numbers corresponding to the logarithms 3.16227, 3.59207, 2.93318, 0.41078, 1.60065, 2.756.86, 3.23528, 1.79692. 4.87800 5.14613.

107. Find the numbers corresponding to the logarithms 3.07372, 3.69205, 1.64904, 2.16107, 0.70364, 1.31952, 4.30814, 3.00087, 2.69949, 6.57978.

108. Find the number corresponding to the logarithms 3.57686, 3.16340, 2.40359, 1.09817, 4.49823, 2.83882, 1.50060, 3.30056, 1.17112, 4.25100.

108. Find the numbers corresponding to the logarithms 3.33720, 3.09875, 0.70093, 4.04640, 2.94004, 1.41509, 2.32649, 4.14631, 3.01290, 5.39003.

Positive logarithms of numbers greater than one are arithmetic sums their characteristics and mantissas. Therefore, operations with them are carried out according to ordinary arithmetic rules.

Negative logarithms of numbers less than one are algebraic sums negative characteristic and positive mantissa. Therefore, operations with them are carried out according to algebraic rules, which are supplemented by special instructions relating to the reduction of negative logarithms to their normal form. Normal form A negative logarithm is one in which the characteristic is a negative integer and the mantissa is a positive proper fraction.

To convert the true reflective logarithm to its normal artificial form, you need to increase absolute value its whole term by one and make the result a negative characteristic; then add all the digits of the fractional term to 9, and the last one to 10 and make the result a positive mantissa. For example, -2.57928 = 3.42072.

To convert the artificial normal form of a logarithm to its true form negative value, you need to reduce the negative characteristic by one and make the result an integer term of the negative sum; then add all the digits of the mantissa to 9, and the last one to 10 and make the result a fractional term of the same negative sum. For example: 4.57406= -3.42594.

109. Convert logarithms to artificial form -2.69537, -4, 21283, -0.54225, -1.68307, -3.53820, -5.89990.

109. Convert logarithms to artificial form -3.21729, -1.73273, -5.42936, -0.51395, -2.43780, -4.22990.

110. Find the true values ​​of logarithms 1.33278, 3.52793, 2.95426, 4.32725, 1.39420, 5.67990.

110. Find the true values ​​of logarithms 2.45438, 1.73977, 3.91243, 5.12912, 2.83770, 4.28990.

The rules for algebraic operations with negative logarithms are expressed as follows:

To apply a negative logarithm in its artificial form, you need to apply the mantissa and subtract the absolute value of the characteristic. If the addition of mantissas produces an integer positive number, then you need to attribute it to the characteristic of the result, making an appropriate amendment to it. For example,

3,89573 + 2 ,78452 = 1 1 ,68025 = 2,68025

1 ,54978 + 2 ,94963=3 1 ,49941=2 ,49941.

To subtract a negative logarithm in its artificial form, you need to subtract the mantissa and add the absolute value of the characteristic. If the subtracted mantissa is large, then you need to make an adjustment in the characteristic of the minuend so as to separate a positive unit from the minuend. For example,

2,53798-3 ,84582=1 1 ,53798-3 ,84582 = 4,69216,

2 ,22689-1 ,64853=3 1 ,22689-1 ,64853=2 ,57836.

To multiply a negative logarithm by a positive integer, you need to multiply its characteristic and mantissa separately. If, when multiplying the mantissa, a whole positive number is identified, then you need to attribute it to the characteristic of the result, making an appropriate amendment to it. For example,

2 ,53729 5=10 2 ,68645=8 ,68645.

When multiplying a negative logarithm by a negative quantity, you must replace the multiplicand with its true value.

To divide a negative logarithm by a positive integer, you need to separate its characteristic and mantissa separately. If the characteristic of the dividend is not exactly divisible by the divisor, then you need to make an amendment to it so as to include several positive units in the mantissa, and make the characteristic a multiple of the divisor. For example,

3 ,79432: 5=5 2 ,79432: 5=1 ,55886.

When dividing a negative logarithm by a negative quantity, you need to replace the dividend with its true value.

Perform the following calculations using logarithmic tables and check the results in the simplest cases using ordinary methods:

174. Determine the volume of a cone whose generatrix is ​​0.9134 feet and whose base radius is 0.04278 feet.

175. Calculate the 15th term of a multiple progression, the first term of which is 2 3 / 5 and the denominator is 1.75.

175. Calculate the first term of a multiple progression, the 11th term of which is equal to 649.5 and the denominator is 1.58.

176. Determine the number of factors A , A 3 , A 5 r . Find something like this A , in which the product of 10 factors is equal to 100.

176. Determine the number of factors. A 2 , A 6 , A 10 ,.... so that their product equals the given number r . Find something like this A , in which the product of 5 factors is equal to 10.

177. The denominator of the multiple progression is 1.075, the sum of its 10 terms is 2017.8. Find the first term.

177. The denominator of the multiple progression is 1.029, the sum of its 20 terms is 8743.7. Find the twentieth term.

178 . Express the number of terms of a multiple progression given the first term A , last and denominator q , and then, randomly choosing numeric values a And u , pick up q so that n

178. Express the number of terms of a multiple progression given the first term A , last And and denominator q And And q , pick up A so that n was some integer.

179. Determine the number of factors so that their product is equal to r . What it must be like r in order to A =0.5 and b =0.9 the number of factors was 10.

179. Determine the number of factors so that their product is equal r . What it must be like r in order to A =0.2 and b =2 the number of factors was 10.

180. Express the number of terms of a multiple progression given the first term A , I'll follow And and the product of all members r , and then, choosing arbitrarily numeric values A And r , pick up And and then the denominator q so that And was some integer.

160. Express the number of terms of a multiple progression given the first term A , the last and and the product of all terms r , and then, randomly selecting numeric values And And r , pick up A and then the denominator q so that n was some integer.

Solve the following equations, where possible - without the help of tables, and where not, with tables:

They often take the number ten. Logarithms of numbers based on base ten are called decimal. When performing calculations with the decimal logarithm, it is common to operate with the sign lg, not log; in this case, the number ten, which determines the base, is not indicated. So, let's replace log 10 105 to simplified lg105; A log 10 2 on lg2.

For decimal logarithms the same features that logarithms have with a base greater than one are typical. Namely, decimal logarithms are characterized exclusively for positive numbers. The decimal logarithms of numbers greater than one are positive, and those of numbers less than one are negative; of two non-negative numbers, the larger one is equivalent to the larger decimal logarithm, etc. Additionally, decimal logarithms have distinctive features and peculiar features that explain why it is comfortable to prefer the number ten as the base of logarithms.

Before examining these properties, let us familiarize ourselves with the following formulations.

Integer part of the decimal logarithm of a number A is called characteristic, and the fractional one is mantissa this logarithm.

Characteristics of the decimal logarithm of a number A is indicated as , and the mantissa as (lg A}.

Let's take, say, log 2 ≈ 0.3010. Accordingly = 0, (log 2) ≈ 0.3010.

Likewise for log 543.1 ≈2.7349. Accordingly, = 2, (log 543.1)≈ 0.7349.

The calculation of decimal logarithms of positive numbers from tables is widely used.

Characteristic features of decimal logarithms.

The first sign of the decimal logarithm. not a whole negative number, represented by a one followed by zeros, is a positive integer equal to the number of zeros in the entry of the selected number .

Let's take log 100 = 2, log 1 00000 = 5.

Generally speaking, if

That A= 10n , from which we get

lg a = lg 10 n = n lg 10 =n.

Second sign. The ten logarithm of a positive decimal, shown as a one with leading zeros, is - n, Where n- the number of zeros in the representation of this number, taking into account zero integers.

Let's consider , log 0.001 = - 3, log 0.000001 = -6.

Generally speaking, if

,

That a= 10-n and it turns out

lga= lg 10n =-n log 10 =-n

Third sign. The characteristic of the decimal logarithm of a non-negative number greater than one is equal to the number of digits in the integer part of this number excluding one.

Let's analyze this feature: 1) The characteristic of the logarithm lg 75.631 is equal to 1.

Indeed, 10< 75,631 < 100. Из этого можно сделать вывод

lg 10< lg 75,631 < lg 100,

1 < lg 75,631 < 2.

It follows from this,

log 75.631 = 1 +b,

Shifting a decimal point in a decimal fraction to the right or left is equivalent to the operation of multiplying this fraction by a power of ten with an integer exponent n(positive or negative). And therefore, when the decimal point in a positive decimal fraction is shifted to the left or right, the mantissa of the decimal logarithm of this fraction does not change.

So, (log 0.0053) = (log 0.53) = (log 0.0000053).