Logarithm is the inverse operation of exponentiation. If you are wondering what power you need to raise 2 to get 10, then the logarithm will come to your aid.

Inverse operation for exponentiation

Exponentiation is repeated multiplication. To raise two to the third power, we need to calculate the expression 2 × 2 × 2. The inverse operation for multiplication is division. If the expression that a × b = c is true, then the inverse expression b = a / c is also true. But how to reverse exponentiation? The multiplication inversion problem has an elegant solution due to simple property, that a × b = b × a. However, a b is not equal to b a , except for the single case that 2 2 = 4 2 . In the expression a b = c, we can express a as the bth root of c, but how do we express b? This is where logarithms come into play.

The concept of the logarithm

Let's try to solve a simple equation like 2 x = 16. This is an exponential equation because we need to find the exponent. For a simpler understanding, let's set the problem like this: how many times do you need to multiply a two by itself in order to get 16 as a result? Obviously, 4, so the root of this equation is x = 4.

Now let's try to solve 2 x = 20. How many times does 2 need to be multiplied by itself to get 20? This is difficult, because 2 4 \u003d 16, and 2 5 \u003d 32. Logically, the root of this equation is located between 4 and 5, and closer to 4, perhaps 4.3? Mathematicians do not tolerate approximate calculations and want to know the exact answer. To do this, they use logarithms, and the root of this equation will be x = log2 20.

The expression log2 20 is read as the logarithm of 20 to base 2. This is the answer, which is enough for strict mathematicians. If you want to express this number exactly, then calculate it using engineering calculator. In this case, log2 20 = 4.32192809489. This is an irrational infinite number, and log2 20 is its compact notation.

In this elegant way, you can solve any simple exponential equation. For example, for equations:

• 4 x = 125, x = log4 125;
• 12 x = 432, x = log12 432;
• 5 x = 25, x = log5 25.

The last answer x = log5 25 mathematicians will not like. This is because log5 25 is easy to calculate and is an integer, so you must define it. How many times does it take to multiply 5 by itself to get 25? Basically, twice. 5 × 5 \u003d 5 2 \u003d 25. Therefore, for an equation of the form 5 x \u003d 25, x \u003d 2.

Decimal logarithm

The decimal logarithm is a base 10 function. It's a popular math tool, so it's written differently. For example, to what power do you need to raise 10 to get 30? The answer would be log10 30, but mathematicians abbreviate decimal logarithms and write it as lg30. Similarly, log10 50 and log10 360 are written as lg50 and lg360, respectively.

natural logarithm

The natural logarithm is a function in base e. There is nothing natural in it, and such a function simply scares many neophytes. The number e = 2.718281828 is a constant that naturally arises when describing processes of continuous growth. How important is the number pi for geometry, the number e plays important role in modeling time processes.

To what power must e be raised to get 10? The answer would be loge 10, but mathematicians denote the natural logarithm as ln, so the answer would be ln10. The same is true for the expressions loge 35 and loge 40, whose correct notation is ln34 and ln40.

Antilog

The antilogarithm is the number that corresponds to the value of the selected logarithm. In simple words, in the expression loga b the antilogarithm is the number b a . For the decimal logarithm lga, the antilogarithm is 10 a , and for the natural lna the antilogarithm is e a . In fact, this is also exponentiation and inverse operation for logarithm.

The physical meaning of the logarithm

Finding degrees - pure mathematical problem, but why do we need logarithms in real life? At the beginning of the development of the idea of ​​​​the logarithm, this mathematical tool was used to reduce volumetric calculations. great physicist and the astronomer Pierre-Simon Laplace said that "the invention of logarithms shortened the work of the astronomer and doubled his life." With the development of a mathematical tool, entire logarithmic tables were created, with the help of which scientists could operate with huge numbers, and the properties of functions make it possible to convert expressions that operate on irrational numbers into integer expressions. Also, logarithmic notation allows you to represent too small and too big numbers in compact form.

Logarithms have also found application in the field of displaying graphic processes. If you want to draw a graph of a function that takes the values ​​1, 10, 1000 and 100000, then the small values ​​will be invisible and visually they will merge into a point near zero. To solve this problem, the decimal logarithm is used, which allows you to plot a function graph that adequately displays all its values.

The physical meaning of the logarithm is a description of temporal processes and changes. For example, the base 2 logarithm allows you to determine how many doublings of the initial value are required to achieve a certain result. The decimal function is used to find the number of decimals needed, and the natural function is the time it takes to reach a given level.

Our program is a collection of four online calculators that allow you to calculate the logarithm to any base, decimal and natural logarithmic function, as well as the decimal antilogarithm. To perform calculations, you will need to enter the base and the number, or just the number for the decimal and natural logarithms.

Real life examples

school task

As mentioned above, irrational values ​​of the type log2 345 do not require additional transformations, and such an answer will completely satisfy the teacher of mathematics. However, if the logarithm is calculated, you must represent it as an integer. Suppose you have solved 5 problems in algebra, and you need to check the results for the possibility of an integer representation. Let's check them with a logarithm calculator to any base:

• log7 65 - irrational number;
• log3 243 - integer 5;
• log5 95 - irrational;
• log8 512 - integer 3;
• log2 2046 - irrational.

So log3 243 and log8 512 would need to be rewritten as 5 and 3 respectively.

Potentiation

Potentiation is finding the antilogarithm of a number. Our calculator allows you to find antilogarithms in base 10, which means raising ten to the power of n. Let's calculate the antilogarithms for the following values ​​of n:

• for n = 1 antlog = 10;
• for n = 1.5 antlog = 31.623;
• for n = 2.71 antlog = 512.861.

Continuous growth

The natural logarithm allows you to describe the processes of continuous growth. Imagine that the GDP of the country of Krakozhia increased from $5.5 billion to $7.8 billion in 10 years. Let's determine the annual GDP growth as a percentage using the natural logarithm calculator. To do this, we need to calculate the natural logarithm of ln(7.8/5.5), which is equivalent to ln(1.418). Let's enter this value into the cell of the calculator and get the result of 0.882 or 88.2% for the entire time. Since GDP has been growing for 10 years, its annual growth will be 88.2 / 10 = 8.82%.

Finding the number of decimals

Let's say that in 30 years the number of personal computers has increased from 250,000 to 1 billion. How many times has the number of PCs increased by 10 times in all this time? To calculate such an interesting parameter, we need to calculate the decimal logarithm lg(1,000,000,000 / 250,000) or lg(4,000). Let's choose a decimal logarithm calculator and calculate its value lg(4,000) = 3.60. It turns out that over time, the number of personal computers has increased 10 times every 8 years and 4 months.

Conclusion

Despite the complexity of logarithms and the dislike of children in their school years, this mathematical tool is widely used in science and statistics. Use our collection of online calculators to solve school assignments, as well as problems from various scientific fields.

The degree of a single number is called mathematical term invented several centuries ago. In geometry and algebra, there are two options - decimal and natural logarithms. They are calculated 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 features

At the moment, there are ten known mathematical qualities. The most common and popular of them are:

• The root log divided by the root value is always the same as the base 10 logarithm √.
• The product of log is always equal to the sum of the producer.
• Lg = the value of the power multiplied by the number that is raised to it.
• If we subtract the divisor from the log dividend, we get lg quotient.

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

Calculating the base 10 logarithm is a rather specific task, so integrating properties into a solution must be approached with care and regularly reviewed for consistency. We must not forget about the tables, with which you need to constantly check, and be guided only by the data found there.

Varieties of a 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 a decimal log. Otherwise, "a" is transformed into "y" and has transcendental and irrational features. It is also worth noting that the natural value is calculated by a special equation, where the theory studied outside the high school curriculum becomes the proof.

Decimal type 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. At the same time, before proceeding directly to the case, you need to build log in In addition, in each store school supplies you can find a special ruler with a printed scale that helps to solve an equation of any complexity.

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

Two kinds of formula

All types and varieties of problems for calculating the answer, which have the term log in the condition, have a separate name and a strict mathematical device. exponential equation is almost an exact copy of the logarithmic calculations, if you look from the side of the correctness of the solution. It's just that the first option includes a specialized number that helps to quickly understand the condition, and the second one replaces log with an ordinary degree. 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 obligatory presence of a base. The standard version of the value is 10. It is marked with the sequence - log x or lg 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 terms is 2.72. The official marking adopted in both school and more complex professional formulas is ln x.
• Different. In addition to the basic logarithms, there are hexadecimal and binary types (base 16 and 2, respectively). There is also the most complicated option with a base indicator of 64, which falls under the systematized control of the adaptive type, which calculates the final result with geometric accuracy.

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

• meaning;
• argument;
• base.

Computing a log number

There are three ways to quickly and verbally do everything necessary calculations to find the result of interest with the obligatory correct outcome of the solution. Initially, we approximate the decimal logarithm to its order (scientific notation of a number in a degree). Each positive value can be given by an equation where it will be equal to the mantissa (a number from 1 to 9) multiplied by ten in nth degree. This calculation option was created on the basis of two mathematical facts:

• the product and sum of 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. The accuracy is improved when 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 there is a specialized table at hand that can be easily used in your evaluation activities. With its help, you can find out what the decimal logarithm is up to tenths of a percent of the original number.

Real log history

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

At the end of the second half of the era, several minds at once 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. In the same way, scientists have integrated and subtraction.

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 huge 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 sign of the index. The closest to understanding the logarithm were the Bernoulli brothers, 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 masses were made at the dawn of the 18th century by Bernoulli and Leibniz. But they failed to compile holistic theoretical calculations. There was a whole discussion about this, but the exact definition of the number was not assigned. Later the dialogue resumed, but between Euler and d'Alembert.

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

tables

The properties of the number indicate that multi-digit numbers can not be multiplied, but found in log 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 searched for in Bradis's 1921 collection. Later, in 1971, the Vega edition appeared.

Often take the number ten. Logarithms of numbers to base ten are called decimal. When performing calculations with the decimal logarithm, it is common to operate with the sign lg, but not log; while the number ten, which determines the base, is not indicated. Yes, we replace log 10 105 to simplified lg105; A log102 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. Decimal logarithms of numbers greater than one are positive, and numbers less than one are negative; of two not negative numbers a larger decimal logarithm is also equivalent to a larger one, etc. Additionally, decimal logarithms have distinctive features and peculiar features, which explain why it is comfortable to prefer the number ten as the basis of logarithms.

Before analyzing these properties, let's take a look at the following formulations.

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

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

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

The same is true for lg 543.1 ≈2.7349. Accordingly, = 2, (lg 543.1)≈ 0.7349.

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

Characteristic signs of decimal logarithms.

The first sign of the decimal logarithm. a non-negative integer represented by 1 followed by zeros is a positive integer equal to the number of zeros in the chosen number .

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

Generally speaking, if

That A= 10n , from which we get

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

Second sign. Decimal logarithm of a positive decimal, shown by a one with leading zeros, is − P, Where P- the number of zeros in the representation of this number, taking into account the zero of integers.

Consider , lg 0.001 = -3, lg 0.000001 = -6.

Generally speaking, if

,

That a= 10-n and it turns out

lga = lg 10n =-n lg 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 equated to 1.

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

lg 10< lg 75,631 < lg 100,

1 < lg 75,631 < 2.

This implies,

lg 75.631 = 1 + b,

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

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

Which is very easy to use, does not require its interface and run any additional programs. All that is required of you is to go to the Google website and enter the appropriate request in the only field on this page. For example, to calculate the base 10 logarithm for 900, enter in the field search query lg 900 and immediately (even without pressing a button) you get 2.95424251.

Use a calculator if you don't have access to a search engine. It can also be a software calculator from the standard set of Windows OS. The easiest way to run it is to press the key combination WIN + R, enter the command calc and click the "OK" button. Another way is to open the menu on the "Start" button and select "All Programs" in it. Then you need to open the "Standard" section and go to the "Utilities" subsection to click the "Calculator" link there. If you are using Windows 7, you can press the WIN key and type "Calculator" in the search field, and then click the corresponding link in the search results.

Switch the interface of the calculator to advanced mode, since the basic version that opens by default does not provide the operation you need. To do this, open the "View" section in the program menu and select the "" or "engineering" item - depending on which version of the operating system is installed on your computer.

At present, you will not surprise anyone with discounts. Sellers understand that discounts are not a means of increasing revenue. The greatest efficiency is not 1-2 discounts for a specific product, but a system of discounts, which should be simple and understandable to the employees of the company and its customers.

Instruction

You probably noticed that at present the most common is growing with an increase in production volumes. In this case, the seller develops a scale of percentage discounts, which increases with the growth of purchases over a certain period. For example, you bought a kettle and coffee maker and received discount 5 %. If you also buy an iron this month, you will receive discount 8% off all purchased items. At the same time, the profit received by the company at a discounted price and increased sales should not be less than the expected profit at a non-discounted price and the same level of sales.

Calculating the scale of discounts is easy. First determine the sales volume at which the discount starts. can be taken as the lower limit. Then calculate the expected amount of profit that you would like to receive on the item you are selling. Its upper limit will be limited by the purchasing power of the product and its competitive properties. Maximum discount can be calculated as follows: (profit - (profit x minimum sales volume / expected volume) / unit price.

Another fairly common discount is the contract discount. This can be a discount on, when buying certain types of goods, as well as when calculating in a particular currency. Sometimes discounts of this plan are provided when buying a product and ordering for delivery. For example, you buy a company's products, order transport from the same company and get discount 5% on purchased goods.

The amount of pre-holiday and seasonal discounts is determined based on the cost of the goods in the warehouse and the probability of selling the goods at a set price. Typically, retailers resort to such discounts, for example, when selling clothes from last season's collections. Such discounts are used by supermarkets in order to unload the work of the store in the evenings and weekends. In this case, the size of the discount is determined by the size of the lost profit in case of non-satisfaction of consumer demand during peak hours.

Sources:

• how to calculate discount percentage in 2019

You may need to calculate logarithms to find values ​​using formulas containing exponents as unknown variables. Two types of logarithms, unlike all the others, have their own names and designations - these are logarithms to bases 10 and the number e (irrational constant). Consider a few simple ways calculating the logarithm to base 10 - the "decimal" logarithm.

Instruction

Use for calculations built in operating system Windows. To run it, press the win key, select the "Run" item in the main menu of the system, enter calc and press OK. The standard interface of this program does not have a function for calculating algorithms, so open the "View" section in its menu (or press the key combination alt + "and") and select the line "scientific" or "engineering".

Welcome to the online logarithm calculator.

What is this calculator for? Well, first of all, in order to check with your written or mental calculations. You can encounter logarithms (in Russian schools) already in the 10th grade. And this topic is considered quite complex. Solving logarithms, especially those with large or fractional numbers You know, it's not easy. It's better to play it safe and use a calculator. When filling out, be careful not to confuse the base with the number. The logarithm calculator is somewhat similar to the factorial calculator, which automatically generates several solutions.
In this calculator, you have to fill in only two fields. Number field and base field. Well, let's try to curb the calculator in practice. For example, you need to find log 2 8 (logarithm of 8 to base 2 or logarithm to base 2 of 8, don't be scared different pronunciations). So, enter 2 in the "Enter base" field, and enter 8 in the "Enter a number" field. Then press "find logarithm" or enter. Next, the logarithm calculator takes the logarithm of the given expression and displays such a result on your screens.

Logarithm calculator (real) - this calculator finds the logarithm to a given base online.
Decimal Logarithm Calculator is a calculator that looks up the base 10 base 10 logarithm online.
Natural logarithm calculator - this calculator that finds the logarithm to the base e online.
Binary Log Calculator is a calculator that finds the base 2 logarithm online.

A bit of theory.

The concept of the real logarithm: There are many different definitions of the logarithm. First, it would be nice to know that the logarithm is some kind of algebraic notation, denoted as log a b, where a is the base, b is a number. And this entry is read like this: Logarithm to the base a of the number b. The notation log b is sometimes used.
The base, that is, "a", is always at the bottom. Since it is always raised to a power.
And now, in fact, the definition of the logarithm itself:
The logarithm of a positive number b to the base a (where a>0, a≠1) is the power to which you need to raise the number a to get the number b. By the way, not only the base must be in a positive form. The number (argument) must also be positive. Otherwise, the logarithm calculator will set off a nasty alarm. Logarithm is the operation of finding the logarithm, given the base. This operation is the inverse of exponentiation with the appropriate base. Compare:

Exponentiation	Logarithm
	log 10 1000 = 3;
	log 03 0.0081=4;

And the operation inverse to logarithm is Potentiation.
In addition to the real logarithm, the base of which can be any number (in addition to negative numbers, zero and one), there are logarithms with a constant base. For example, the decimal logarithm.
The base 10 logarithm of a number is the base 10 logarithm, which is written as lg6, or lg14. It looks like a spelling mistake or even a typo in which the Latin letter "o" is missing.
The natural logarithm is the base logarithm equal to the number e, for example ln7, ln9, e≈2.7. There is also the binary logarithm, which is not as important in mathematics as it is in information theory and computer science. The base of the binary logarithm is 2. For example: log 2 10.
Decimal and natural logarithms have the same properties as the logarithms of numbers with any positive base.