Logarithms

In mathematics, logarithms are inverse functions to exponents . That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. 

In the simplest case, the logarithm counts repeated multiplication of the same factor; e.g., since 1000 = 10 × 10 × 10 = 103, the "logarithm to base 10" of 1000 is 3. The logarithm of x to base b is denoted as logb (x) (or, without parentheses, as logbx, or even without explicit base as log x, when no confusion is possible). 

More generally, exponents allow any positive real number to be raised to any real power, always producing a positive result, so the logarithm for any two positive real numbers b and x where b is not equal to 1, is always a unique real number y. More explicitly, the defining relation between exponentiation and logarithm is:

   exactly if   


For example, log2 64 = 6, as 64 = 26.

The 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 (that is b ≈ 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.

Plots of logarithm functions of three commonly used bases. The special points logb b = 1 are indicated by dotted lines, and all curves intersect in logb 1 = 0. image: wikipedia


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:

logb (xy) = logb x + logb y

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.

Where are logarithmic scales used?

Logarithmic scales reduce wide-ranging quantities to tiny scopes. For example, the decibel (dB) is a unit used to express ratio as logarithms, mostly for signal power and amplitude (of which sound pressure is a common example). In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They help describing frequencyratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, and can aid in forensic accounting.

In the same way as logarithms reverses exponents, the complex logarithm is the inverse function of exponential functions applied to complex numbers. The discrete logarithm is another variant; it has uses in public-key cryptography.

Source adapted from:Wikipedia contributors. (2019, February 7). Logarithm. In Wikipedia, The Free Encyclopedia. Retrieved 07:26, February 12, 2019, from https://en.wikipedia.org/w/index.php?title=Logarithm&oldid=882257893


Commentary:

Note: the wikipedia page talks a lot about exponentiation which is pretty uncommon usage wording so here we use exponents in the plural in the same way as logarithms is pluralised.

Here is an excellent video tutorial on how to deal with logarithms