Common logarithm;Briggs logarithm)，Also calledDecimal logarithm, refers to the base 10 logarithm.The common logarithm of positive number x is lgx.It is composed of Napier andBriggsProposed.At first, they worked together to compile the decimal logarithm table, which was finally completed by Briggs in 1624, so it is also calledBriggs logarithm。The popular logarithmic table is evolved from Briggs logarithmic table.The common logarithm of a number can be written as the sum of an integer and a positive number less than 1, for example, lgb=n+lgN (n is an integer, 1 ≤ N<10), where the integer part n is called the first number of the logarithm, and the positive decimal part lgN is called the mantissa.The integer part of the common logarithm of a number greater than 1 is the number of digits before the decimal point minus 1.For a number less than 1, if there are P zeros after the decimal point, the first number of its logarithm is p-1.For example, in lg 200=2.3010, 2 is the first number and 0.3010 is the mantissa, while in lg 0.02=- 2+0.3010, the first number is - 2 and the mantissa is+0.3010.Common logarithm hasNatural logarithmAdvantages not available: if one positive number is 10 times the other, the common logarithm will increase by 1, and so on[1]。
Common logarithm, also called "decimal logarithm", is an important mathematical tool, which is based on 10logarithm。The common logarithm of positive number N can be recorded as
, often omitting the base number of 10
。The common logarithm of any positive number can be written as an integer（positive integer, zeronegtive integer）In the form of a positive pure decimal (or zero), the integral part is called the common logarithmiccharacteristicThe part of the positive pure decimal (or zero) is called the common logarithmicMantissa。for example
, the first number is
, the approximate value of mantissa is
；
, the first number is
, the approximate value of mantissa is
。Before the invention of the computer, the logarithm with the base of 10 was a common tool in complex numerical calculation, so it was called the common logarithm.Briggs(H. Briggs) first proposed to improve the logarithm to a common logarithm based on 10 for easy calculation.In memory of him, the common logarithm is also namedBriggs logarithm。
Properties of common logarithms
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In addition to general logarithmic properties, common logarithms have the following special properties:
1. If
, then
It is a positive pure decimal.
2. If
(n is an integer), then
。
3. If
(n is an integer,
）, then
Pure decimal (or zero), where the integer part n is called the first number, and pure decimal (or zero) is called the mantissa.
4. The logarithmic mantissa of two numbers not less than 1 with different decimal points is the same, for example, according to the four digit logarithmic table,
The mantissa of is
。
5. The first logarithm of a number not less than 1 isNonnegative integer, which is equal to the number of digits in front of the decimal point of the number minus 1, for example
The first number of is 2;The first logarithm of a number between zero and 1 is a negative integer. Its absolute value is equal to the number of consecutive zeros after the decimal point of the number plus 1, for example
The first number of is
，
The first number of is
。
6. The first and last numbers of commonly used logarithms are often written togetherdecimal pointSeparation, e.g
In this writing method, the mantissa is always a positive pure decimal;The first number is an integer, which can be positive, negative or zero.[2]。
