What is a logarithm? Exact definition sounds so: "number A logarithm on the basis of C call an exponent in which it is necessary to build number C to receive number A". In the standard record it looks so: log with A. Naprimer, a logarithm 8 on the basis 2 is equal 3, and the logarithm 256 on the same basis is equal to 8.
If the logarithm basis (that is, that number which needs to be built in degree) is 10, then the logarithm is called "decimal", and is designated as follows: lg. If as the basis transcendental number e acts (approximately equal 2.718), then the logarithm is called "natural", and ln is designated. For what in general logarithms are necessary? What from them practical advantage? Perhaps, the famous mathematician, the physicist and the astronomer Pierre-Simone Laplace best of all answered these questions (1749-1827). According to him, the invention of such indicator as a logarithm as if doubles life of astronomers, reducing calculations of several months in work of several days. Some can answer it: like, it is a little fans of mysteries of the star sky, and to other people what logarithms give? Speaking about astronomers, Laplace meant, first of all, those who are engaged in difficult calculations. And the invention of logarithms very much facilitated this work. In the Middle Ages of the mathematician in Europe, as well as many other sciences, practically did not develop. It happened, first of all, because of domination of the church which was jealously watching that the scientific word did not disperse from the Scripture. But gradually, with growth of number of the universities and also with the invention of the press of the mathematician began to revive. The strongest impetus in development of discipline was given by an era of Great Geographical Discoveries. The sailors sailing on search of new lands needed both accurate maps, and astronomical tables for positioning of the ship. And for their drawing up the combined efforts of astronomers observers and mathematicians calculators were required. The special merit in this association belongs to the ingenious scientist, Johann Kepler (1571 - 1630) who made fundamental discoveries, working on the theory of celestial motion. He made astronomical tables very exact (for those times). But the calculations necessary for their drawing up, still remained very difficult, they demanded enormous efforts and big expenses of time. And so proceeded until logarithms were invented. With their help became possible many times over to simplify and accelerate calculations. Using the tables of logarithms made by the famous Scottish mathematician John Napier it is possible to multiply with little effort numbers, to take roots. The logarithm allows to simplify multiplication of multidigit numbers by addition of their logarithms. For example, we will take two numbers which need to be increased by means of logarithms: 45.2 and 378. By means of the table we will see that on the basis the 10th these numbers are equal 1.6551 and 2.5775, that is, 45.2 =10^1.6551 and 378=10^2.5775. Thus, 45.2*378=10^(1.6551+2.5775)=10^4.2326. Received that the logarithm of the work of numbers 45.2 and 378 is equal to 4.2326. From the table of logarithms it is easy to find result of the work.