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In mathematics, the logarithm is the inverse function to exponentiation. That means that the logarithm of a number x to the base b is the exponent to which b must be raised to produce x. For example, since 1000 = 103, the logarithm base of 1000 is 3, or log10 (1000) = 3.
The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718 281 828 459. [1] The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x.
ln (r) is the standard natural logarithm of the real number r. Arg (z) is the principal value of the arg function; its value is restricted to (−π, π]. It can be computed using Arg (x + iy) = atan2 (y, x). Log (z) is the principal value of the complex logarithm function and has imaginary part in the range (−π, π].
He then called the logarithm, with this number as base, the natural logarithm. As noted by Howard Eves, "One of the anomalies in the history of mathematics is the fact that logarithms were discovered before exponents were in use." Carl B. Boyer wrote, "Euler was among the first to treat logarithms as exponents, in the manner now so familiar."
Common logarithm. A graph of the common logarithm of numbers from 0.1 to 100. In mathematics, the common logarithm is the logarithm with base 10. [1] It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered its use, as well as ...
Iterated logarithm. In computer science, the iterated logarithm of , written log * (usually read " log star "), is the number of times the logarithm function must be iteratively applied before the result is less than or equal to . [1] The simplest formal definition is the result of this recurrence relation :
Binary logarithm. Graph of x as a function of a positive real number x. In mathematics, the binary logarithm ( log2 n) is the power to which the number 2 must be raised to obtain the value n. That is, for any real number x , For example, the binary logarithm of 1 is 0, the binary logarithm of 2 is 1, the binary logarithm of 4 is 2, and the ...
In mathematics, the logarithmic integral function or integral logarithm li ( x) is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a very good approximation to the prime-counting function, which is defined as the number of prime numbers ...