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The principal argument of any positive real number x is 0; hence Log(x) is a real number and equals the real (natural) logarithm. However, the above formulas for logarithms of products and powers do not generalize to the principal value of the complex logarithm. [96]
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].
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, log e x, or sometimes, if the base e is implicit, simply log x.
On the real line, the gamma function has a local minimum at z min ≈ +1.46163 21449 68362 34126 [15] where it attains the value Γ(z min) ≈ +0.88560 31944 10888 70027. [16] The gamma function rises to either side of this minimum. The solution to Γ(z − 0.5) = Γ(z + 0.5) is z = +1.5 and the common value is Γ(1) = Γ(2) = +1.
The prime number theorem then states that x / log x is a good approximation to π(x) (where log here means the natural logarithm), in the sense that the limit of the quotient of the two functions π(x) and x / log x as x increases without bound is 1:
This value can then be used to give some scaling relation between the inflexion point and maximum point of the log-normal distribution. [55] This relationship is determined by the base of natural logarithm, e = 2.718 … {\displaystyle e=2.718\ldots } , and exhibits some geometrical similarity to the minimal surface energy principle.
The area of the blue region converges to Euler's constant. Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (γ), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log:
Later elements up to 10,000,000 of the same sequence a n = log(n) − n/ π (n) (red line) appear to be consistently less than 1.08366 (blue line). Legendre's constant is a mathematical constant occurring in a formula constructed by Adrien-Marie Legendre to approximate the behavior of the prime-counting function π ( x ) {\displaystyle \pi (x)} .