Logarithm

                                                       

                                       Logarithm

In mathematics, the logarithm of a number is the exponent to which another fixed value, the base, must be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 10 to the power 3 is 1000 (as1000 = 10 × 10 × 10 = 10³). More precisely, for any two positive real numbers b and x where b is not equal to 1, the logarithm of x to base b, denoted log_b(x), is the unique real number y such that
b^y = x.

for example, as 64 = 4³, we havelog₄(64) = 3The logarithm to base 10 (that is b = 10) is called the common logarithm and has many applications in science and engineering. The natural logarithm has the number e (≈ 2.718) as its base; its use is widespread in mathematics and physics, because of its simpler derivative. The binary logarithm uses base 2 (that is b = 2) and is commonly used in computer science.
Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. They were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. Tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition because of the fact — important in its own right — that the logarithm of a product is the sum of the logarithms of the factors:

provided that b, x and y are all positive and b ≠ 1. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century.
Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the decibel is a unit quantifying signal power log-ratios and amplitude log-ratios (of which sound pressure is a common example). In chemistry, pH is a  logarithmic measure for the acidity of anaqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They describe musical intervals, appear in formulas counting prime numbers, inform some models inpsychophysics, and can aid in forensic accounting.In the same way as the logarithm reverses exponentiation, the complex logarithm is the inverse function of the exponential function applied to complex numbers. The discrete logarithm is another variant; it has uses in public-key cryptography.

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