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The natural logarithm of e itself, ln e, is 1, because e 1 = e, while the natural logarithm of 1 is 0, since e 0 = 1. The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/x from 1 to a [4] (with the area being negative when 0 < a < 1). The simplicity of this definition, which is matched in many ...
In mathematics, a polylogarithmic function in n is a polynomial in the logarithm of n, [1] () + () + + () +.The notation log k n is often used as a shorthand for (log n) k, analogous to sin 2 θ for (sin θ) 2.
These are the three main logarithm laws/rules/principles, [3] from which the other properties listed above can be proven. Each of these logarithm properties correspond to their respective exponent law, and their derivations/proofs will hinge on those facts. There are multiple ways to derive/prove each logarithm law – this is just one possible ...
The definition for the gamma function due to Weierstrass is also valid for all complex numbers except non-positive integers: = = (+) /, where is the Euler–Mascheroni constant. [1] This is the Hadamard product of 1 / Γ ( z ) {\displaystyle 1/\Gamma (z)} in a rewritten form.
The logarithm is denoted "log b x" (pronounced as "the logarithm of x to base b", "the base-b logarithm of x", or most commonly "the log, base b, of x "). An equivalent and more succinct definition is that the function log b is the inverse function to the function x ↦ b x {\displaystyle x\mapsto b^{x}} .
A common unit of information is the bit or shannon, based on the binary logarithm. Other units include the nat, which is based on the natural logarithm, and the decimal digit, which is based on the common logarithm. In what follows, an expression of the form p log p is considered by convention to be equal to zero whenever p = 0.
In mathematics, the common logarithm (aka "standard logarithm") is the logarithm with base 10. [1] It is also known as the decadic logarithm , the decimal logarithm and the Briggsian logarithm . The name "Briggsian logarithm" is in honor of the British mathematician Henry Briggs who conceived of and developed the values for the "common logarithm".
Since the common logarithm of a power of 10 is exactly the exponent, the characteristic is an integer number, which makes the common logarithm exceptionally useful in dealing with decimal numbers. For positive numbers less than 1, the characteristic makes the resulting logarithm negative, as required. [38]