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 spelling are always equal to each other. This identity characterizes properties that relate to the useful potential of the function.
Features and important features
At the moment, ten well-known mathematical qualities are distinguished. The most common and sought after 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 magnitude of the degree multiplied by the number that is raised in it.
- If we take away the divisor from the log of the dividend, we get the lg quotient.
In addition, there is an equation based on the main identity (considered key), a transition to an updated base and several secondary formulas.
The calculation of the decimal logarithm is a rather specific task, therefore, it is necessary to approach the integration of properties into the solution carefully and regularly check your actions and sequence. We must not forget about the tables, which must 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 score of 10, then this is a decimal log. Otherwise, “a” transforms into “y” and has transcendental and irrational attributes. It is also worth noting that the natural quantity is calculated by a special equation, where the theory studied outside the high school curriculum becomes proof.
Decimal logarithms are widely used in the calculation of complex formulas. Entire tables have been compiled that facilitate calculations and clearly show the process of solving the problem. In this case, before going straight to the point, you need to put log in a standard form. In addition, in each school supplies store 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 Briggov, or Euler's digit, in honor of the researcher who was the first to publish the quantity and find the contrast between the two definitions.
Two kinds of formulas
All types and varieties of tasks for calculating the answer, which have the term log in the terms, have a separate name and a strict mathematical structure. The exponential equation is an almost exact copy of the logarithmic calculations, when viewed from the side of the correctness of the solution. Just the first option includes a specialized number that helps you quickly figure out the condition, 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 characteristics that distinguish 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 log x.
- Natural. If its base is the “e” sign, which is a constant identical to a strictly calculated equation, where n is rapidly moving to infinity, then the approximate size of the number in digital equivalent is 2.72. The official marking adopted in both school and more complex professional formulas is ln x.
- Different. In addition to the main logarithms, hexadecimal and binary types are found (base 16 and 2, respectively). There is still the most difficult option with a base indicator of 64, which falls under the systematic control of the adaptive type, calculating the final result with geometric accuracy.
The terminology includes the following quantities included in the algebraic problem:
Calculation of 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 decision. Initially, we bring the decimal logarithm to its order (scientific notation of a number in a power). Each positive value can be set by the equation, where it will be equal to the mantissa (digit from 1 to 9), multiplied by ten in the nth power. This version of the calculation is based on two mathematical facts:
- the product and the 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.
- If an error in the calculation nevertheless occurs, then it is never less than one in the direction of subtraction.
- The accuracy is improved when you consider that lg with a base of three has a final result - five tenths of a unit. Therefore, any mathematical value greater than 3 automatically adds one point to the answer.
- Almost perfect accuracy is achieved if you have at hand a specialized table that can be easily applied in your assessment activities. With its help, you can find out what the decimal logarithm of up to tenths of the original number is equal to.
Real log history
The sixteenth century acutely felt the need for more complex calculi than was known to the science of that time. This was especially true for the division and multiplication of multivalued digits 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 juxtaposed two
progressions: arithmetic and geometric. In this case, all basic calculations should rest against the latter value. In the same way, scientists integrated subtraction.
The first mention of lg took place in 1614. This was done by an amateur mathematician named Neper. It is worth noting that, despite the huge popularization of the results, a mistake 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 Euler was the first to legitimize the debut in the eighteenth century. He also extended the function to the field of education.
History of Integrated Log
Debut attempts to integrate lg into the masses were made at the dawn of the 18th century by Bernoulli and Leibniz. But they did not manage to compose integral theoretical calculations. A whole discussion was held on this subject, but no exact number was assigned. The dialogue later resumed, but between Euler and D'Alembert.
The latter, in principle, agreed with many facts proposed by the founder of the quantity, 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 the number indicate that multivalued digits can not be multiplied, but to find their log and add them through specialized tables.
This indicator became especially valuable 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, the publication of Vega appeared.