User:WillowW/Logarithm

Original lead (15 Feb 2011)


The logarithm of a number to a given base is the exponent to which the base must be raised in order to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 10 to the power of 3 is 1000: 103 = 1000. The logarithm of x to the base b is written as logb(x), such as log10(1000) = 3.

Via the following formulas, logarithms reduce products to sums:
 * $$ \log_b(xy) = \log_b x + \log_b y \,$$

and powers to products:
 * $$ \log_b(x^p) = p \log_b x. \,$$

John Napier invented logarithms in the early 17th century. Before calculators became available in the latter half of the 20th century, logarithm tables and slide rules greatly simplified scientific calculations. For such computational purposes, the logarithm to base b = 10 (common logarithm) was primarily used. The natural logarithm uses base b = e. It is critical to calculus since it is the inverse function of the exponential function. The binary logarithm uses base b = 2 and primarily aids computing applications.

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the Richter scale is the common logarithm of the amplitude of a seismic event. Logarithms are commonplace in scientific formulas, measure the complexity of algorithms and of fractals, and appear in formulas counting prime numbers. They describe musical intervals, inform some models in psychophysics and can aid in forensic accounting.

The complex logarithm is the inverse of the exponential function applied to complex numbers and generalizes the logarithm to complex numbers. The discrete logarithm is anouther variant; it has applications in public-key cryptography.

Suggestions for lead
In mathematics, the logarithm of a number x equals the number of times a given base b must be multiplied to give x. For example, if x equals m powers of b (x = bm), then m is the logarithm of x to the base b; this is written m = logbx. Hence, the logarithm is the inverse of exponentiation to the same base b


 * x = blogbx

and similarly


 * m = logb bm

The logarithm of zero is undefined, regardless of the base, since there is no well-defined number m such that bm = 0. (Are there exceptions? Do we need to handle the b=0 case more explicitly?)

Although the base b is arbitrary, commonly chosen bases include 2 (the binary logarithm), 10 (the common logarithm) and e (the natural logarithm). The common logarithm log10x counts the powers of ten (i.e., the number of decades) required to reach a number x. The binary logarithm log2x counts the powers of 2 required to reach x; this number is often measured in decibels. Finally, the natural logarithm ln x counts the powers of e required to reach x. Logarithms in any given base a can be converted to logarithms in any other base b using the equation


 * logbx = logax logba

Logarithms have several advantages. First, a wide variation of numbers (a large dynamic range) can be represented by a small range of logarithms. For illustration, the million-fold variation from 1 to 106 can be represented by common logarithms between 0 and 6. Large variations of numbers can be presented graphically on logarithmic scales. Examples include the log-log plot and the semi-log plot, which render power laws and exponential decays as straight lines. Second, logarithms reduce the operation of multiplication to an equivalent addition. The product of two numbers x = bm and y = bn can be written as


 * x · y = bm · bn = bm+n

The equivalent equation in logarithms is


 * logb (x · y) = logb (bm+n) = m+n = logb (x) + logb (y)

Thus, the logarithm of a product x·y equals the sum of the logarithms of the individual factors, logb (x) + logb (y). This equation forms the basis for multiplying two numbers on a slide rule. Since adding two numbers is generally easier than multiplying two numbers, this led to the rapid adoption of logarithms for calculations after their invention by John Napier in the early 17th century.

Logarithms have found ubiquitous applications in science and mathematics, particularly those that feature large numbers and repeated multiplications. Examples include...don't forget pH and activated reaction rates

Logarithms can be thought of as a logarithmic function, defined as the inverse of the corresponding exponentiation function for the same base. discuss complex numbers here

''Perhaps work the following sentences/worked example into the introductory paragraph above? Hmmm, too long and distracting? It might be better in a section.'' The logarithm counts the number of steps in a geometric progression with common ratio b, that is, the number of b-fold increases required to obtain an x-fold increase. As an illustration, how often must a population double before it has increased 32-fold? The answer is five, because 32 = 25 = 2×2×2×2×2; this answer is provided directly by logarithms: log232 = 5.

Thoughts about the 2nd version
[Probably we should move all this to Talk:Logarithm.]

Thanks, Willow, for the draft. A few comments: Jakob.scholbach (talk) 20:25, 16 February 2011 (UTC)
 * How can we explain the concept of exponent in the lead? It would be great if we can come up with a two-liner explaining it. Currently, it is pretty much implicit in the 10^3 = 1000 example. Somehow we should evoke the "number of multiplications", yet we should not convey the impression that it only works for integer exponents. The latter might be emphasized in the first section, though.
 * I would not mention the inverse function relation so early/prominently; I don't think this is digestible even for many people who already know logarithms. Also, let us put only those formulas that are absolutely necessary (which I think does not include $$b^{log_b(x)} = x$$)
 * The product formula is IMO the one and most important property of log's. Otherwise they probably wouldn't even exist, both from the historical and the mathematical point of view. Reducing the scale is secondary, I believe.
 * The lead should stay focused. The current one is relatively short, so a bit longer might be OK, but details on log-log graphs, say, seem to be out of place to me. Also, mentioning that log(0) is undefined is not necessary here, I believe.
 * I like this sentence: "This equation forms the basis for multiplying two numbers on a slide rule. Since adding two numbers is generally easier than multiplying two numbers, this led to the rapid adoption of logarithms for calculations after their invention by John Napier in the early 17th century."