Decimal logarithm: how to calculate?

The degree of a particular number is called a mathematical term, invented several centuries ago. In geometry and algebra, there are two variants - decimal and natural logarithms. They are calculated by different formulas, while the equations differing in spelling are always equal to each other. This identity characterizes the properties that relate to the useful potential of the function.

Features and Signs

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 decimal logarithm.
• The product log is always equal to the sum of the producer.
• Lg = the value of the degree multiplied by the number that is built into it.
• If we divide the divisor from the log divisor, we obtain lg of the quotient.

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

Calculating the decimal logarithm is a very specific task, therefore it is necessary to approach the integration of properties in the solution carefully and regularly check their actions and consistency. We should not forget about tables that need to be constantly checked, and be guided only by the data found there.

Varieties of the mathematical term

The main differences of the mathematical number are "hidden" in the base (a). If it has a value of 10, then it is a decimal log. In the opposite case, "a" is transformed into "y" and has transcendental and irrational signs. It is also worth noting that the natural quantity is calculated by a special equation, where the proof is the theory studied outside the school program of the senior classes.

Logarithms of the decimal type are widely used in the calculation of complex formulas. Complete tables are made that facilitate calculations and clearly demonstrate the process of solving the problem. In this case, before proceeding directly to the case, you need to raise log to the standard view. In addition, in each store of school supplies, you can find a special ruler with a scale that helps to solve an equation of any complexity. The decimal logarithm of a number is called Brigg's, or Euler's figure, in honor of the researcher who first published the value and found the opposition of the two definitions.

Two types of formula

All types and types 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, if viewed from the correctness of the solution. Simply the first option includes a specialized number that helps you understand the condition more quickly, and the second replaces log with an ordinary degree. In this case, calculations using the last formula should include a variable value.

Difference and terminology

Both main indicators have their own peculiarities that distinguish numbers from each other:

• The decimal logarithm. An important detail of the number is the obligatory presence of a foundation. 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 the strictly calculated equation, where n rapidly moves to infinity, then the approximate size of the number in the digital equivalent is 2.72. The official marking adopted in both school and in more complex professional formulas is ln x.
• Different. In addition to the basic logarithms, hexadecimal and binary species occur (bases 16 and 2, respectively). There is still the most complicated variant with a base index of 64, falling under the systematized control of the adaptive type, with geometrical accuracy, which produces the calculation of the final result.

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

• value;
• argument;
• base.

Calculating the log number

There are three ways to quickly and verbally do all the necessary calculations to find the desired result with the required correct outcome of the solution. Initially, we approximate the decimal logarithm to its order (the scientific notation of a number in degree). Each positive value can be given by an equation where it will be equal to the mantissa (a digit from 1 to 9) multiplied by a ten in the n-th power. Such a calculation is based on two mathematical facts:

• The product and the sum of log always have the same value;
• The logarithm taken from a number from one to ten can not exceed a value of 1 point.

1. If the error in the calculation still occurs, then it is never less than one in the direction of subtraction.
2. Accuracy increases if we consider that lg with base three has a final result - five tenths of one. Therefore, any mathematical value greater than 3 automatically adds one item to the answer.
3. Practically perfect accuracy is achieved if there is a specialized table at hand, which can be easily applied in its evaluation actions. With its help you can find out what the decimal logarithm is to ten percent of the original number.

The history of the real log

The sixteenth century acutely felt the need for more complex calculi than was known to the science of the time. Especially it concerned the division and multiplication of many-valued digits with a large sequence, including fractions. At the end of the second half of the era, several minds immediately came to the conclusion that numbers were added using a table that compared two progressions: arithmetic and geometric. At the same time, all the basic calculations had to rest on the last value. In the same way, scientists have integrated and subtracted.

The first mention of lg was in 1614. This was done by a mathematician mate named Nepper. It is worth noting that, despite the huge popularization of the results obtained, the formula was made error due to ignorance of some definitions that appeared later. It began with the sixth sign of the indicator. The closest to understanding 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

The debut attempts to integrate lg into the broad masses were made at the dawn of the 18th century by Bernoulli and Leibniz. But they did not manage to compile complete theoretical calculations. On this subject, a whole discussion was held, but the exact determination of the number was not appropriated. Later the dialogue resumed, but already 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 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 a number indicate that multivalued digits can not be multiplied, but they can be found and logged by means of specialized tables.

Especially valuable this indicator has become for astronomers, who are forced to work with a large set of sequences. In Soviet times, the decimal logarithm was searched in the collection of Bradis, released in 1921. Later, in 1971, appeared the publication of Vega.