Number x logarithm on the basis of an is called such number y that a^y = x. As logarithms facilitate very many practical calculations, it is important to be able to use them.
1. Number x logarithm on the basis of a we will designate loga(x). For example, log2(8) is number 8 logarithm on the basis 2. It is equal 3 because 2^3 = 8.
2. The logarithm is defined only for positive numbers. Negative numbers and zero have no logarithms regardless of the basis. At the same time the logarithm can be any number.
3. Any positive number, except unit can form the basis of a logarithm. However in practice two bases are most often used. Logarithms on the basis 10 are called decimal and are designated by LG (x). Decimal logarithms most often meet in practical calculations.
4. The second popular basis for logarithms — irrational transcendental number e = 2.71828 … The logarithm on the basis of e is called natural and ln(x) is designated. The e^x and ln (x) functions possess the special characteristics important for differential and integral calculus therefore natural logarithms are more often used in the mathematical analysis.
5. The logarithm of the work of two numbers is equal to the sum of logarithms of these numbers on the same basis: loga(x*y) = loga(x) + loga(y). For example, log2(256) = log2(32) + log2(8) = 8. The logarithm of private two numbers is equal to the difference of their logarithms: loga (x/y) = loga(x) - loga(y).
6. To find a logarithm of the number built in degree it is necessary to increase a logarithm of the number by an exponent: loga(x^n) = n*loga(x). At the same time the exponent can be any number — positive, negative, zero, whole or fractional. As x^0 = 1 for any x, loga(1) = 0 for any a.
7. The logarithm replaces multiplication with addition, exponentiation by multiplication, and extraction of a root division. Therefore for lack of computer facilities the logarithmic tables considerably simplify calculations. To find a logarithm of the number which is not entering the table it it is necessary to present in the form of the work of two or more numbers which logarithms are in the table and to find final result, having put these logarithms.
8. Rather easy way to calculate a natural logarithm — to use decomposition of this function in power series: ln (1 + x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 + … + ((-1) ^ (n + 1)) * (/n). a row gives (x^n) ln values (1 + x) for-1 <x ≤1. In other words, it is so possible to calculate natural logarithms of numbers from 0 (but not including 0) up to 2. Natural logarithms of numbers outside this row can be found by summation of found, using that the logarithm of the work is equal to the sum of logarithms. In particular ln(2x) = ln(x) + ln (2).
9. For practical calculations it is sometimes convenient to pass from natural logarithms to decimal. Any transition from one basis of logarithms to another is made on a formula: logb(x) = loga(x)/loga(b). Thus, log10(x) = ln(x)/ln(10).