The degree of a single number is called a mathematical term 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 writing 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 sought after of them are:
- The radical log divided by the root value is always the same as the decimal 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 to be key), the transition to the updated base andsome minor formulas.
Calculating the base 10 logarithm is a rather specific task, so integrating properties into a solution must be done carefully and regularly check your steps 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 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 logarithms are widely used in calculating complex formulas. Entire tables have been compiled to facilitate calculations and clearly show the process of solving the problem. In this case, before proceeding directly to the case, you need to raise the log to a standard form. In addition, in every school supply store you can find a special ruler with a printed scale that helps you solve an equation of any complexity.
The decimal logarithm of a number is called Brigg's, or Euler's digit, after the researcher who first published the value and discovered the opposition between the two definitions.
Two kinds of formula
All types andvarieties of problems for calculating the answer, which have the term log in the condition, have a separate name and a strict mathematical device. The exponential equation is almost an exact copy of the logarithmic calculations, when viewed 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. However, 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 an adaptive type, which calculates the final result with geometric accuracy.
The terminology includes the following quantities included in the algebraictask:
- value;
- argument;
- base.
Calculate 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 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 to the nth power. This calculation option is based on two mathematical facts:
- product and sum log always have the same exponent;
- logarithm taken from a number from one to ten cannot exceed 1 point.
- If an error in the calculation does occur, then it is never less than one in the direction of subtraction.
- The accuracy improves 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.
- Almost perfect accuracy is achieved if you have a specialized table at hand that you can easily use in your evaluation activities. With its help, you can find out what the decimal logarithm is equal to tenth percent of the original number.
History of real log
The sixteenth century was in dire need of more complex calculus than was known to the science of the time. Especially thisconcerned 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 adding numbers using a table that compared two progressions: arithmetic and geometric. 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 inas the final version. In addition, Euler published the derivative of the decimal logarithm and compiled the first graphs.
Tables
Number properties indicate that multi-digit numbers can not be multiplied, but found 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 times, the decimal logarithm was searched for in the collection of Bradis, released in 1921. Later, in 1971, the Vega edition appeared.
