1, logarithmic algorithm: a log (a) n = n (a > 0 and a is not equal to1)) log (a) n = n (a >; 0 and a are not equal to 1) log (a) Mn = log (a) m+log (a) n (a >: 0 a is not equal to1). log(a)M/N = log(a)M-log(a)N(a >； 0 a is not equal to 1). log(a)^m^n=nlog(a)^m(a>； 0 a is not equal to 1).

2. Operational nature of logarithmic function: if a=em, then m is called logarithm of e with a base, logea=m, e is the base of natural logarithm, and it is an infinite acyclic decimal, which is defined as follows: if an = b(A>；; 0, a is not equal to 1), then n=logea.

In mathematics, logarithm is the reciprocal of power, just as division is the reciprocal of multiplication, and vice versa. This means that the logarithm of a number is an exponent that must produce another fixed number (radix). In a simple example, the logarithmic count factor in the multiplier.

More generally, the power operation allows any positive real number to be raised to any power, and always produces positive results, so the logarithm of any two positive real numbers b and x whose b is not equal to 1 can be calculated.

Application of Logarithm:

Logarithm has many applications both inside and outside mathematics. Some of these events are related to the concept of scale invariance. For example, each chamber of the Nautilus shell is a rough copy of the next chamber, scaled by a constant factor. This leads to a logarithmic spiral. Benford's law about the distribution of pre-derivatives can also be explained by scale invariance. Logarithm is also related to self-similarity.

For example, the logarithmic algorithm appears in the algorithm analysis, and the algorithm is decomposed into two similar smaller problems, and their solutions are patched, and the problem is solved. The size of self-similar geometric shapes, that is, shapes whose parts are similar to the whole image, is also based on logarithm. Logarithmic scale is useful for quantifying the relative change of value relative to its absolute difference.

In addition, because the logarithmic function log(x) grows very slowly for larger x, the logarithmic scale is used to compress large-scale scientific data. Logarithm also appears in many scientific formulas, such as tsiolkovsky rocket equation, Fenske equation or Nernst equation.