Decimal logarithm online. Logarithm

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

Reverse operation for exponentiation

Exponentiation is repeated multiplication. To raise two to the third power, we need to evaluate 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 do we convert the exponentiation? The multiplication reversal problem has an elegant solution thanks to simple property that a × b = b × a. However, a b is not equal to b a, except in the only case where 2 2 = 4 2. In the expression a b = c, we can express a as the bth root of c, but how to express b? This is where logarithms come into the picture.

The concept of 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 pose the problem like this: how many times do you need to multiply two by itself in order to get 16 as a result? Obviously 4, so the root given equation x = 4.

Now let's try to solve 2 x = 20. How many times do we need to multiply two by itself to get 20? This is difficult, because 2 4 = 16, and 2 5 = 32. Logically, the root of this equation is located between 4 and 5, and closer to 4, perhaps 4.3? Mathematicians hate approximate calculations and want to know the exact answer. This is why they use logarithms, and the root of this equation is x = log2 20.

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

You can solve any simple exponential equation in this elegant way. For example, for the equations:

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

Mathematicians will not like the last answer x = log5 25. This is because log5 25 is easy to calculate and is an integer, so you must determine it. How many times does 5 need to be multiplied by itself to get 25? Elementary, twice. 5 × 5 = 5 2 = 25. Therefore, for an equation of the form 5 x = 25, x = 2.

Decimal logarithm

Decimal logarithm is a function in base 10. This is a popular mathematical tool, so it is written differently. For example, to what power should you raise 10 to get 30? The answer would be log10 30, but mathematicians shorten the notation for decimal logarithms and write it as log30. Similarly, log10 50 and log10 360 are written as log50 and log360 respectively.

Natural logarithm

The natural logarithm is a function of base e. There is nothing natural about it, and many neophytes are simply frightened by this function. The number e = 2.718281828 is a constant that naturally arises when describing continuous growth processes. How important is the number Pi for geometry, the number e plays important role in modeling of 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 written as ln10. The same thing applies to the expressions loge 35 and loge 40, the correct form of 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 lga, the antilogarithm is equal to 10 a, and for the natural logarithm lna, the antilogarithm is equal to e a. In fact, this is also exponentiation and the inverse operation for logarithmization.

Physical meaning of the logarithm

Finding powers is a purely mathematical problem, but what are logarithms used for? real life? At the beginning of the development of the idea of ​​logarithms, this mathematical tool was used to reduce voluminous calculations. Great physicist and the astronomer Pierre-Simon Laplace said that “the invention of logarithms shortened the astronomer’s work and doubled his life.” With the development of mathematical tools, 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 transform expressions operating with irrational numbers into integer expressions. Also, logarithmic notation allows you to represent too small and too big numbers in a compact form.

Logarithms have also found application in the field of depicting graphic processes. If you want to draw a graph of a function that takes values ​​1, 10, 1,000 and 100,000, then 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 build a graph of a function that adequately displays all its values.

The physical meaning of logarithm is a description of temporary processes and changes. Thus, 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 tenfolds required, and the natural function represents 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 ​​like log2 345 do not require additional transformations, and such an answer will completely satisfy a mathematics teacher. However, if the logarithm is being 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 integer representation. Let's check them using a logarithm calculator in any base:

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

Thus, you will need to rewrite the values ​​of log3 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 to the decimal base, which literally means raising ten to the power 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 us to describe processes of continuous growth. Let's imagine that the GDP of the country of Krakozhia increased from 5.5 billion dollars to 7.8 in 10 years. Let's determine the annual percentage growth of GDP using the natural logarithm calculator. To do this, we need to calculate the natural logarithm ln(7.8/5.5), which is equivalent to ln(1.418). Let's enter this value into the calculator cell and get the result 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 over 30 years the number personal computers 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 log(4,000) = 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 children’s dislike for them in school years, this mathematical tool finds wide application in Science and Statistics. Use our collection of online calculators to solve school assignments, as well as problems from various scientific fields.

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 being 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 signs

On this 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 between a 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 widely used in the calculation of complex formulas. Entire tables have been compiled to facilitate calculations and clearly show the process of solving the problem. In this case, before directly getting down to business, you need to raise log to In addition, in every school supply store you can find a special ruler with a printed scale that helps solve an equation of any complexity.

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

Two types of formula

All types and varieties of problems for calculating the answer, having the term log in the condition, have a separate name and a strict mathematical structure. Exponential equation is 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 own characteristics, distinguishing numbers from each other:

• Decimal logarithm. An important detail of the number is the mandatory presence of a base. The standard version of the value is 10. It is marked with the sequence - log x or log x.
• Natural. If its base is the sign "e", which is a constant identical to a strictly calculated equation, where n rapidly moves towards infinity, then the approximate size of the number in digital equivalent is 2.72. The official marking, adopted both in school and in more complex professional formulas, is ln x.
• Different. In addition to basic logarithms, there are hexadecimal and binary types (base 16 and 2, respectively). There is an even more complex option with a base indicator of 64, which falls under 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 do everything necessary calculations to find the result of interest with the obligatory correct outcome of the decision. 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 closest to the understanding of the logarithm were the Bernoulli brothers, and the debut legalization occurred in the eighteenth century by Euler. He also extended the function to the field of education.

History of complex log

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

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

Tables

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

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

Welcome to the online logarithm calculator.

What is this calculator used for? Well, first of all, in order to check your written or mental calculations. You can come across 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 an easy matter. 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 produces several solutions.
In this calculator, you only have to fill out two fields. A field for a number and a field for a base. Well, let's try to harness 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 number” field. Then press “find logarithm” or enter. Next, the logarithm calculator logarithms the given expression and displays the following result on your screens.

Logarithm (real) calculator – this calculator finds the logarithm using a given base online.
Decimal Logarithm Calculator is a calculator that looks up the base 10 decimal logarithm online.
Calculator natural logarithms- this calculator that searches for the logarithm to base e online.
Binary Logarithm Calculator is a calculator that finds base 2 logarithms online.

Concept of real logarithm: There are many different definitions of logarithm. First, it would be nice to know that a logarithm is a kind of algebraic notation, denoted as log a b, where a is the base and b is a number. And this entry reads like this: Logarithm to base a of 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:
Logarithm positive number b to base a (where a>0, a≠1) is the power to which the number a must be raised to obtain the number b. By the way, not only the base must be in positive form. The number (argument) must also be positive. Otherwise, the logarithm calculator will trigger an unpleasant alarm. Logarithm is the operation of finding a logarithm based on a given base. This operation is the inverse of exponentiation with the corresponding base. Compare:

And the inverse operation of logarithm is Potentiation.
In addition to the real logarithm, the base of which can be any number (besides negative numbers, zero and one), there are logarithms with a constant base. For example, the decimal logarithm.
The decimal logarithm of a number is a logarithm to base 10, written as lg6, or lg14. It looks like a spelling error or even a typo in which the Latin letter “o” is missing.
A natural logarithm is a logarithm with a base 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 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 of numbers with any positive base.

Which is very easy to use, does not require any additional programs to be installed in its interface. All you have to do is go to the Google website 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 query field and immediately (even without pressing a button) you will get 2.95424251.

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

Switch the calculator interface 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 “ ” or “engineering” - depending on the version of the operating system installed on your computer.

Nowadays you won't surprise anyone with discounts. Sellers understand that discounts are not a means of increasing revenue. The most effective is not 1-2 discounts on a specific product, but a system of discounts that should be simple and understandable to the company’s employees and its customers.

Instructions

You have probably noticed that currently the most common one is growing with increasing production volumes. IN in this case the seller develops a scale of discount percentages, which increases with the growth of purchase volumes 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% on all purchased goods. At the same time, the company's profit received at a discounted price and increased sales volume should be no less than the expected profit at a price without a discount and the same sales level.

Calculating the discount scale is easy. First, determine the sales volume from which the discount begins. You can take as a lower limit. Then calculate the expected amount of profit you would like to make on the product you sell. 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 the contract discount. This may be a discount upon purchase certain types goods, as well as when paying in one currency or another. Sometimes discounts of this type are provided when purchasing goods and ordering for delivery. For example, you buy a company’s products, order transport from the same company and receive 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 likelihood of selling the goods at the set price. Typically, retailers resort to such discounts, for example, when selling clothes from last season's collections. Supermarkets use similar discounts in order to relieve the workload of the store in evening hours and weekends. In this case, the size of the discount is determined by the amount of lost profits when consumer demand is not satisfied during peak hours.

Sources:

• how to calculate the discount percentage in 2019

Calculating logarithms may be necessary to find values ​​using formulas containing exponents as unknown variables. Two types of logarithms, unlike all others, have their own names and notations - these are logarithms to bases 10 and the number e (an irrational constant). Let's look at a few simple ways calculating the base 10 logarithm - the "decimal" logarithm.

Instructions

Use for calculations built into the Windows operating system. To run it, press the win key, select “Run” in the main menu of the system, enter calc and click 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 alt + “and” key combination) and select the “scientific” or “engineering” line.

So, we have powers of two. If you take the number from the bottom line, you can easily find the power to which you will have to raise two to get this number. For example, to get 16, you need to raise two to the fourth power. And to get 64, you need to raise two to the sixth power. This can be seen from the table.

And now - actually, the definition of the logarithm:

The base a logarithm of x is the power to which a must be raised to get x.

Designation: log a x = b, where a is the base, x is the argument, b is what the logarithm is actually equal to.

For example, 2 3 = 8 ⇒ log 2 8 = 3 (the base 2 logarithm of 8 is three because 2 3 = 8). With the same success log 2 64 = 6, since 2 6 = 64.

The operation of finding the logarithm of a number to a given base is called logarithmization. So, let's add a new line to our table:

Unfortunately, not all logarithms are calculated so easily. For example, try finding log 2 5 . The number 5 is not in the table, but logic dictates that the logarithm will lie somewhere on the segment. Because 2 2< 5 < 2 3 , а чем больше степень двойки, тем больше получится число.

Such numbers are called irrational: the numbers after the decimal point can be written ad infinitum, and they are never repeated. If the logarithm turns out to be irrational, it is better to leave it that way: log 2 5, log 3 8, log 5 100.

It is important to understand that a logarithm is an expression with two variables (the base and the argument). At first, many people confuse where the basis is and where the argument is. To avoid annoying misunderstandings, just look at the picture:

Before us is nothing more than the definition of a logarithm. Remember: logarithm is a power, into which the base must be built in order to obtain an argument. It is the base that is raised to a power - it is highlighted in red in the picture. It turns out that the base is always at the bottom! I tell my students this wonderful rule at the very first lesson - and no confusion arises.

We've figured out the definition - all that remains is to learn how to count logarithms, i.e. get rid of the "log" sign. To begin with, we note that two important facts follow from the definition:

1. The argument and the base must always be greater than zero. This follows from the definition of a degree by a rational exponent, to which the definition of a logarithm is reduced.
2. The base must be different from one, since one to any degree still remains one. Because of this, the question “to what power must one be raised to get two” is meaningless. There is no such degree!

Such restrictions are called range of acceptable values(ODZ). It turns out that the ODZ of the logarithm looks like this: log a x = b ⇒ x > 0, a > 0, a ≠ 1.

Note that there are no restrictions on the number b (the value of the logarithm). For example, the logarithm may well be negative: log 2 0.5 = −1, because 0.5 = 2 −1.

However, now we are considering only numerical expressions, where it is not required to know the VA of the logarithm. All restrictions have already been taken into account by the authors of the problems. But when they go logarithmic equations and inequalities, DHS requirements will become mandatory. After all, the basis and argument may contain very strong constructions that do not necessarily correspond to the above restrictions.

Now let's consider general scheme calculating logarithms. It consists of three steps:

1. Express the base a and the argument x as a power with the minimum possible base greater than one. Along the way, it’s better to get rid of decimals;
2. Solve the equation for variable b: x = a b ;
3. The resulting number b will be the answer.

That's all! If the logarithm turns out to be irrational, this will be visible already in the first step. The requirement that the base be greater than one is very important: this reduces the likelihood of error and greatly simplifies the calculations. Same with decimals: if you immediately convert them to regular ones, there will be many fewer errors.

Let's see how this scheme works using specific examples:

Task. Calculate the logarithm: log 5 25

1. Let's imagine the base and argument as a power of five: 5 = 5 1 ; 25 = 5 2 ;

Let's create and solve the equation:
log 5 25 = b ⇒ (5 1) b = 5 2 ⇒ 5 b = 5 2 ⇒ b = 2 ;

2. We received the answer: 2.

Task. Calculate the logarithm:

Task. Calculate the logarithm: log 4 64

1. Let's imagine the base and argument as a power of two: 4 = 2 2 ; 64 = 2 6 ;
2. Let's create and solve the equation:
   log 4 64 = b ⇒ (2 2) b = 2 6 ⇒ 2 2b = 2 6 ⇒ 2b = 6 ⇒ b = 3 ;
3. We received the answer: 3.

Task. Calculate the logarithm: log 16 1

1. Let's imagine the base and argument as a power of two: 16 = 2 4 ; 1 = 2 0 ;
2. Let's create and solve the equation:
   log 16 1 = b ⇒ (2 4) b = 2 0 ⇒ 2 4b = 2 0 ⇒ 4b = 0 ⇒ b = 0 ;
3. We received the answer: 0.

Task. Calculate the logarithm: log 7 14

1. Let's imagine the base and argument as a power of seven: 7 = 7 1 ; 14 cannot be represented as a power of seven, since 7 1< 14 < 7 2 ;
2. From the previous paragraph it follows that the logarithm does not count;
3. The answer is no change: log 7 14.

A small note on the last example. How can you be sure that a number is not an exact power of another number? It's very simple - just break it down into prime factors. If the expansion has at least two different factors, the number is not an exact power.

Task. Find out whether the numbers are exact powers: 8; 48; 81; 35; 14 .

8 = 2 · 2 · 2 = 2 3 - exact degree, because there is only one multiplier;
48 = 6 · 8 = 3 · 2 · 2 · 2 · 2 = 3 · 2 4 - is not an exact power, since there are two factors: 3 and 2;
81 = 9 · 9 = 3 · 3 · 3 · 3 = 3 4 - exact degree;
35 = 7 · 5 - again not an exact power;
14 = 7 · 2 - again not an exact degree;

Note also that the prime numbers themselves are always exact powers of themselves.

Decimal logarithm

Some logarithms are so common that they have a special name and symbol.

The decimal logarithm of x is the logarithm to base 10, i.e. The power to which the number 10 must be raised to obtain the number x. Designation: lg x.

For example, log 10 = 1; log 100 = 2; lg 1000 = 3 - etc.

From now on, when a phrase like “Find lg 0.01” appears in a textbook, know that this is not a typo. This is a decimal logarithm. However, if you are unfamiliar with this notation, you can always rewrite it:
log x = log 10 x

Everything that is true for ordinary logarithms is also true for decimal logarithms.

Natural logarithm

There is another logarithm that has its own designation. In some ways, it's even more important than decimal. It's about about the natural logarithm.

The natural logarithm of x is the logarithm to base e, i.e. the power to which the number e must be raised to obtain the number x. Designation: ln x .

Many will ask: what is the number e? This is an irrational number, its exact value impossible to find and record. I will give only the first figures:
e = 2.718281828459...

We will not go into detail about what this number is and why it is needed. Just remember that e is the base of the natural logarithm:
ln x = log e x

Thus ln e = 1 ; ln e 2 = 2; ln e 16 = 16 - etc. On the other hand, ln 2 is an irrational number. In general, the natural logarithm of any rational number irrational. Except, of course, for one: ln 1 = 0.

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