Questions tagged [logarithm]

A base-b logarithm of a number represents the exponent that b must be raised to to obtain the original number. Namely, for a number n, it is like asking the question "b to which power equals n?"

Given a number n

• Express n = b^x for some base b, and some exponent x.
• Then, the base-b logarithm of n is the exponent x. log_b(n) = x.

The three most important bases of a logarithm is the base-2 logarithm, prevalent in computer science, the base-10 logarithm similar to counting the number of decimal places, and the natural (base-e) logarithm with unique mathematical properties.

The notations of the logarithm of a number n for different bases is

• log_b(n) for a given base b
• log(n), which may mean the base-10 logarithm, the natural logarithm, or a generic logarithm
• ln(n), which is often but not always used for the natural logarithm.

Furthermore, the base of the logarithm can be changed using a simple formula involving a ratio of logarithms. In fact, logarithms of base c is expressed as a constant multiple of a logarithm of base b.

• log_c(n) = ^log_b(n)/_logb(c)

Most of the time, the logarithm is an irrational number that cannot be expressed exactly in terms of decimals, let alone floating-point numbers.

Logarithm in programming languages

Unlike simple additive and multiplicative operators, and in some cases, the exponent, using the logarithm for a given programming language is usually used as a particular built-in library method. The logarithm usually accepts floating-point inputs strictly greater than zero and returns the approximate logarithm of the number, usually the natural logarithm.

Furthermore, note that the name log(...) often refers to the natural logarithm in the context of programming languages, rather than ln(...). The base-10 log is usually named log10(...).

Languages which support a built-in base-b logarithm passes through two arguments, and may either be of the form log(x, b) or log(b, x) for a number to be evaluated x and base b.

Usages in various programming languages (base-e)

• C: log(double x), logf(float x), logl(long double x) (#include <math.h>)
• C++: std::log(x) (#include <cmath>)
• Java: java.lang.Math.log(double x)
• JavaScript: Math.log(x) (do not confuse with console.log)
• Python: log(x), log(x, b) for base-b logarithm (import log from math)
• Rust: ln(self: f64) -> f64, (log10 and log2 for bases 10, 2)
• Fortran: LOG(x)
• MySQL: LOG(x), LOG(b, x) for base-b logarithm of a number x
• Excel: LN(x), LOG(x, b) for base-b

Applications of logarithms

The logarithm is the inverse of exponentiation, namely that log_b(bⁿ), b^log_b(n) are identities that equal the original number n.

Logarithms are important in many mathematical fields, especially those involving variables differing by many orders of magnitude, logarithmic axes, and solving formulas involving exponential expressions. It is also a way to interpolate the number of digits (of some base b) of a given number: the number of decimal digits required for a positive integer n is ⌊log₁₀(n)⌋.

The logarithm also represents a major complexity class in algorithmic complexity theory. Logarithmic complexity represents time or space complexity of O(log(n)) for an input of size n. Logarithmic growth is very slow, and is asymptotically slower than any power function n^c where c is strictly positive including the linear function. Therefore, logarithmic complexity is very efficient, and is considered to be in polynomial time.

These algorithms are all logarithmic-time:

• Binary Search
• BST Insertion (average case)

There is also another major complexity class called linearithmic complexity, which represents O(n * log(n)) complexity. It usually occurs when a logarithmic-complexity process is executed n times. It is asymptotically slower than any power function n^c where c > 1.

These algorithms are all in linearithmic, or n-log-n time.

• Quicksort (average case)
• Listing all elements in a BST (average case)

Read more

Documentation

• C
• C++
• Java
• JavaScript
• Python 2, Python 3
• Rust
• Fortran
• MySQL
• Excel

Tags

• log: DO NOT use this tag for logarithms. The word "log" is also used to refer to the action of logging. That tag is for logarithms and logarithmic concepts only.
• ln: DO NOT use this tag for the natural logarithm. That refers to the link command. Instead, use natural-logarithm.
• exponentiation, pow: Inverses of the logarithm function
• exp: Refers to the natural exponent, the inverse of log(...).
  Namely, log(exp(x)) == exp(log(x)) == x for x > 0.
• complexity, time-complexity, space-complexity, asymptotic-complexity: The logarithm is an important class of algorithmic complexity. It is very efficient, much more efficient than linear, but is lesser than constant.

External links

• Wikipedia for Logarithms