Taking logarithm is the opposite of exponentiation. If you are wondering how much you need to raise 2 to get 10, then the logarithm will come to your rescue.
Reverse 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 \u003d c is true, then the reverse expression b \u003d a / c is also true. But how do you reverse exponentiation? The problem of inverse multiplication has an elegant solution due to the simple property that a × b \u003d b × a. However, a b does not equal b a, except for the only case where 2 2 \u003d 4 2. In the expression a b \u003d c, we can express a as the bth root of c, but how do we express b? This is where logarithms come in.
Logarithm concept
Let's try to solve a simple equation like 2 x \u003d 16. This is an exponential equation, since we want to find the exponent. For a simpler understanding, let's set the problem as follows: how many times do you need to multiply two by itself in order to get 16 as a result? Obviously, 4, so the root of this equation is x \u003d 4.
Now let's try to solve 2 x \u003d 20. How many times do you need to multiply 2 by itself to get 20? This is difficult, because 2 4 \u003d 16, and 2 5 \u003d 32. Logically speaking, the root of this equation is located between 4 and 5, and closer to 4, perhaps 4.3? Mathematicians don't tolerate rough calculations and want to know the exact answer. For this, they use logarithms, and the root of this equation will be x \u003d log2 20.
The expression log2 20 reads logarithm base 2 of 20. This is the answer that is enough for strict mathematicians. If you want to express this number exactly, then calculate it using an engineering calculator. In this case log2 20 \u003d 4.32192809489. This is an irrational infinite number, and log2 20 is its compact record.
In this elegant way, you can solve any simple exponential equation. For example, for equations:
- 4 x \u003d 125, x \u003d log4 125;
- 12 x \u003d 432, x \u003d log12 432;
- 5 x \u003d 25, x \u003d log5 25.
The last answer x \u003d log5 25 will not be pleasant to mathematicians. 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? Elementary, two times. 5 × 5 \u003d 5 2 \u003d 25. Therefore, for an equation like 5 x \u003d 25, x \u003d 2.
Decimal logarithm
The decimal logarithm is a base 10 function. It is a popular mathematical tool, so it is written differently. For example, to what degree does 10 need to be raised to get 30? The answer would be log10 30, but mathematicians abbreviate the logarithm decimal notation 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 base e function. There is nothing natural in it, and many neophytes are simply afraid of this function. The number e \u003d 2.718281828 is a constant that naturally arises when describing processes of continuous growth. Just as Pi is important to geometry, e plays an important role in modeling temporal processes.
To what power should the number e be raised to get 10? The answer would be loge 10, but mathematicians refer to the natural logarithm as ln, so the answer would be written as ln10. It is the same with the expressions loge 35 and loge 40, the correct notation for which is ln34 and ln40.
Antilog
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 of lga, the antilogarithm is 10 a, and for the natural lna, the antilogarithm is e a. In fact, this is also exponentiation and the inverse operation for logarithm.
The physical meaning of the logarithm
Finding degrees is a purely mathematical task, but what are logarithms for in real life? At the beginning of the development of the idea of \u200b\u200blogarithm, this mathematical tool was used to reduce the volume of calculations. The great physicist and astronomer Pierre-Simon Laplace said that "the invention of logarithms reduced the astronomer's work and doubled his life." With the development of a mathematical tool, whole logarithmic tables were created, with the help of which scientists could operate with huge numbers, and the properties of functions allow converting expressions operating with irrational numbers into integer expressions. Also, the logarithmic notation allows you to represent too small and too large numbers in a compact form.
Logarithms have also found application in the field of graphical processes. If you want to draw a graph of a function that takes the values \u200b\u200b1, 10, 1,000 and 100,000, then small values \u200b\u200bwill not be visible and visually they will merge to a point near zero. To solve this problem, the decimal logarithm is used, which allows you to build a graph of a function that adequately displays all of its values.
The physical meaning of logarithm is a description of temporal processes and changes. So, logarithm base 2 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 required decimal places, 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, and decimal antilogarithm. To perform calculations, you will need to enter a base and a number, or just a number for decimal and natural logarithms.
Real life examples
School task
As mentioned above, irrational values \u200b\u200blike log2 345 do not require additional transformations, and this 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 examples in algebra, and you need to check the results for the possibility of an integer representation. Let's check them out with the logarithm calculator for any base:
- log7 65 - irrational number;
- log3 243 - integer 5;
- log5 95 - irrational;
- log8 512 - integer 3;
- log2 2046 - irrational.
Thus, you need to rewrite the values \u200b\u200blog3 243 and log8 512 as 5 and 3, respectively.
Potentiation
Potentiation is finding the antilogarithm of a number. Our calculator allows you to find antilogarithms in decimal base, which means raising ten to the power of n. Let's calculate antilogarithms for the following n values:
- for n \u003d 1 antlog \u003d 10;
- for n \u003d 1.5 antlog \u003d 31.623;
- for n \u003d 2.71 antlog \u003d 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 percentage growth of GDP using the natural log calculator. To do this, we need to calculate the natural logarithm ln (7.8 / 5.5), which is equivalent to ln (1.418). We enter this value into the cell of the calculator and get the result 0.882 or 88.2% for the entire time. Since the GDP has been growing for 10 years, its annual growth will be 88.2 / 10 \u003d 8.82%.
Finding the number of tenths
Let's say 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 10 times over 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) \u003d 3.60. It turns out that over time, the number of personal computers increased 10 times every 8 years and 4 months.
Conclusion
Despite the complexity of logarithms and the dislike of children for them during 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 different scientific fields.
Welcome to the online logarithm calculator.
What is this calculator for? Well, first of all, in order to check your written or mental calculations. Logarithms (in Russian schools) can be encountered already in the 10th grade. And this topic is considered quite complex. Solving logarithms, especially with large or fractional numbers, you know, is not easy. Better to play it safe and use a calculator. When filling in, 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 (log base 2 of 8 or log base 2 of 8, don't be intimidated by the different pronunciation). So, we enter 2 in the field "enter the base", and 8 we enter in the field "enter the number". Then we press "find the logarithm" or enter. Next, the logarithm calculator will logarithm the given expression and display this result on your screens.
Logarithm calculator (real) - this calculator finds the logarithm of a given base online.
Decimal logarithm calculator is a calculator that searches for decimal logarithm with base 10 online.
Natural logarithm calculator - This calculator that searches the base e logarithm online.
Binary Logarithm Calculator is a calculator that finds logarithm base 2 online.
| A bit of theory. |
Real logarithm concept: 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 record reads like this: Logarithm base a of b. Sometimes the notation is log b.
The base, that is, "a" is always below. 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\u003e 0, a ≠ 1) is the degree to which the number a must be raised to get the number b. By the way, not only the foundation should be in a positive form. The number (argument) must also be positive. Otherwise, the logarithm calculator will trigger an unpleasant alarm. Logarithm is the operation of finding the logarithm, given a base. This operation is the reverse of exponentiation with an appropriate radix. Compare:
Exponentiation |
Logarithm |
log 10 1000 \u003d 3; |
|
log 03 0.0081 \u003d 4; |
And the operation inverse to the 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, decimal logarithm.
The decimal logarithm of a number is the base 10 logarithm, written lg6, or lg14. It looks like a spelling mistake or even a typo with the Latin letter "o" 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 logarithms for numbers with any positive base.
Which is very easy to use, does not require in its interface and run any additional programs. All you have to do is go to the Google site and enter the appropriate query in the only field on this page. For example, to calculate the decimal logarithm for 900, enter lg 900 in the search field and immediately (even without clicking the button) you will 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 Windows operating system. The easiest way to launch it is to press the WIN + R key combination, enter the calc command, and click the OK button. Another way is to expand the menu on the Start button and select All Programs. Then you need to open the "Standard" section and go to the "Service" subsection to click the "Calculator" link there. For Windows 7, you can press the WIN key and type calculator in the search box, and then click the related link in the search results.
Switch the calculator interface to advanced mode, as 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 "" or "engineering" - depending on which version of the operating system is installed on your computer.
Nowadays you won't surprise anyone with discounts. Sellers understand that discounts are not revenue-raising. The most effective is not 1-2 discounts for a specific product, but a system of discounts, which should be simple and understandable for the company's employees and its customers.
Instructions
You have probably noticed that currently the most common is growing with increasing production volumes. In this case, the seller develops a scale of discount percentages, which increases with an increase in the volume of purchases for a certain period. For example, you bought a kettle and a coffee maker and received discount five %. If you also buy an iron this month, you will receive discount 8% on all purchased items. At the same time, the company's profit received at a discount price and increased sales volume should be no less than the expected profit at a discount price and the previous sales level.
It is not difficult to calculate the scale of discounts. First, determine the sales volume from which the discount starts. You can take as the lower limit. Then calculate the expected amount of profit that you would like to receive on the product being sold. 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 / expected volume) / unit price.
Another fairly common discount is contract discount. This can be a discount on, when purchasing certain types of goods, as well as when calculating in a particular currency. Sometimes discounts of such a plan are provided when purchasing a product and ordering for delivery. For example, you buy the 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 goods in stock and the likelihood of selling the goods at a set price. Typically, these discounts are used by retailers, for example, when selling clothes from last season's collections. Such discounts are used by supermarkets in order to relieve the work of the store in the evenings and weekends. In this case, the size of the discount is determined by the amount of lost profits in case of dissatisfaction with consumer demand during peak hours.
Sources:
- how to calculate discount percentage in 2019
Calculation of logarithms may be needed to find values \u200b\u200busing 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 of 10 and the number e (an irrational constant). Let's look at some simple ways to calculate the base 10 logarithm - the “decimal” logarithm.
Instructions
Use for computing built into the Windows operating system. To run it, press the win key, select Run from the main system menu, enter calc and press OK. In the standard interface of this program there is no function for calculating algorithms, so open the "View" section in its menu (or press the alt + "and" key combination) and select the line "scientific" or "engineering".
The power of a given number is called a mathematical term that was coined several 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 properties that relate to the useful potential of a function.
Features and important signs
At the moment, there are ten known mathematical qualities. The most common and demanded ones are:
- The root log divided by the root value is always the same as the decimal logarithm √.
- The log product is always equal to the manufacturer's sum.
- Lg \u003d the value of the power multiplied by the number that is raised to it.
- If you subtract the divisor from the log of the dividend, you get lg of the quotient.
In addition, there is an equation based on the main identity (considered key), a transition to an updated radix, and a few minor formulas.
Calculating the decimal logarithm is a rather specific task, so you need to be careful when integrating properties into a solution and regularly check your actions and 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 decimal log. In the opposite case, "a" is transformed into "y" and has transcendental and irrational signs. It is also worth noting that the actual value is calculated by a special equation, where the theory studied outside the high school curriculum becomes the proof.
Decimal logarithms are widely used when calculating 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 matter, you need to build log in. In addition, in every school supply store, you can find a special ruler with a marked scale that helps to solve an equation of any complexity.
The decimal logarithm of a number is called Brigg's or Euler's number, after 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, having the term log in the condition, have a separate name and a strict mathematical device. The exponential equation is practically an exact copy of the logarithmic calculations when viewed from the point of view 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 numbers from each other:
- Decimal logarithm. An important detail of the number is the mandatory presence of a base. The standard variant 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 mark, adopted in both school and more complex professional formulas, is ln x.
- Various. In addition to the basic logarithms, there are hexadecimal and binary types (base 16 and 2, respectively). There is an even more 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:
- value;
- argument;
- base.
Calculating a 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 result of the decision. Initially, we bring the decimal logarithm closer to our order (scientific notation of a number in power). Each positive value can be specified by an equation, where it will be equal to the mantissa (a number from 1 to 9) multiplied by ten to the nth power. This calculation option is based on two mathematical facts:
- the product and the sum of log always have the same indicator;
- 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 towards the subtraction.
- Accuracy is improved when you consider that a base three lg has a final result of five tenths of one. Therefore, any mathematical value greater than 3 automatically adds one point to the answer.
- Almost ideal accuracy is achieved if there is a specialized table at hand, which can be easily used in your evaluative actions. With its help, you can find out what the decimal logarithm is equal to tenths of a percent of the original number.
Real log history
The sixteenth century was acutely in need of more complex calculus than was known to the science of that time. This was especially true of the division and multiplication of 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 the addition of numbers using a table that compared two and a geometric one. Moreover, all basic calculations had to rest against 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 digit of the indicator. The Bernoulli brothers were closest to understanding the logarithm, and the debut legalization took place in the eighteenth century by Euler. He also extended the function to the field of education.
Complex log history
Bernoulli and Leibniz made their debut attempts to integrate lg into the general public at the dawn of the 18th century. But they did not manage to draw up integral theoretical calculations. There was a whole discussion about this, but the exact definition of the number was not assigned. Later, the dialogue was resumed, but this time between Euler and D'Alembert.
The latter agreed in principle 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 they can be found log and added using specialized tables.
This indicator has become especially valuable for astronomers who have to work with a large set of sequences. In Soviet times, the decimal logarithm was looked for in the collection of Bradis, published in 1921. Later, in 1971, the Vega edition appeared.
Often they take the number ten. Logarithms of numbers to base ten name decimal... When performing calculations with the decimal logarithm, it is generally accepted to operate with the sign lg, but not log; however, the number ten defining the base is not indicated. So, we replace log 10 105 to a simplified lg105; and log 10 2 on lg2.
For decimal logarithms typical are the same features that have logarithms with a base greater than one. 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 non-negative numbers, the larger is also equivalent to the larger decimal logarithm, etc. Additionally, decimal logarithms have distinctive features and peculiar features, which explain why it is convenient 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 andreferred to characteristic, and fractional - mantissaof this logarithm.
Characteristic of the decimal logarithm of a number andis indicated as, and the mantissa as (lg and}.
Let us take, say, log 2 ≈ 0.3010, respectively \u003d 0, (log 2) ≈ 0.3010.
Similarly for lg 543.1 ≈2.7349. Accordingly, \u003d 2, (log 543.1) ≈ 0.7349.
The calculation of decimal logarithms of positive numbers using tables is quite common.
Signs of decimal logarithms.
The first sign of the decimal logarithm.a non-negative integer, represented by one followed by zeros, is a positive integer equal to the number of zeros in the record of the selected number .
Take, lg 100 \u003d 2, lg 1 00000 \u003d 5.
Generalized if
Then and= 10 n , from which we get
lg a \u003d lg 10 n \u003d n lg 10 \u003dp.
Second sign. The decimal logarithm of a positive decimal, shown by one followed by zeros, is - pwhere p - the number of zeros in the representation of this number, including zero integers.
Consider , lg 0.001 \u003d - 3, lg 0.000001 \u003d -6.
Generalized if
,
Then a= 10 -n and it turns out
lga \u003d lg 10 n \u003d -n lg 10 \u003d -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 \u003d 1 + b,
Shifting a decimal point to the right or left is equivalent to multiplying this fraction by a power of ten with an integer p(positive or negative). And therefore, when the comma 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, (lg 0.0053) \u003d (lg 0.53) \u003d (lg 0.0000053).