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Natural logarithm

exponentiation: ln ⁡ 1 = 0 , ln ⁡ e = 1 , ln ⁡ ( x y ) = ln ⁡ x + ln ⁡ y for x > 0 and y > 0 , ln ⁡ ( x / y ) = ln ⁡ x − ln ⁡ y for x > 0 and y > 0 , ln ⁡ ( x

Look up ln in Wiktionary, the free dictionary. LN, Ln or ln may refer to: Lawful Neutral, an alignment in Dungeons & Dragons The Sims 3: Late Night, the

Gamma distribution

is ln x. The information entropy is H ⁡ ( X ) = E ⁡ [ − ln ⁡ p ( X ) ] = E ⁡ [ − α ln ⁡ β + ln ⁡ Γ ( α ) − ( α − 1 ) ln ⁡ X + β X ] = α − ln ⁡ β + ln ⁡

Ln (Unix)

ln is a shell command for creating a link file to an existing file or directory. By default, the command creates a hard link, but with the -s command line

Stirling's approximation

function: ln ⁡ n ! = ln ⁡ 1 + ln ⁡ 2 + ⋯ + ln ⁡ n . {\displaystyle \ln n!=\ln 1+\ln 2+\cdots +\ln n.} The right-hand side of this equation minus 1 2 ( ln ⁡ 1

Natural logarithm of 2

bases is obtained with the formula log b ⁡ 2 = ln ⁡ 2 ln ⁡ b . {\displaystyle \log _{b}2={\frac {\ln 2}{\ln b}}.} The common logarithm in particular is (OEIS: A007524)

Beta distribution

G X ) ( ln ⁡ ( 1 − X ) − ln ⁡ G 1 − X ) ] = E ⁡ [ ( ln ⁡ X − E ⁡ [ ln ⁡ X ] ) ( ln ⁡ ( 1 − X ) − E ⁡ [ ln ⁡ ( 1 − X ) ] ) ] = E ⁡ [ ln ⁡ X ln ⁡ ( 1 −

Log-normal distribution

X [ ln ⁡ X ≤ ln ⁡ x ] = d d x Φ ( ln ⁡ x − μ σ ) = φ ( ln ⁡ x − μ σ ) d d x ( ln ⁡ x − μ σ ) = φ ( ln ⁡ x − μ σ ) 1 σ x = 1 x σ 2 π exp ⁡ ( − ( ln ⁡ x

Softplus

{\displaystyle x} it is ln ⁡ ( 1 + e x ) = ln ⁡ ( 1 + ϵ ) ⪆ ln ⁡ 1 = 0 {\displaystyle \ln(1+e^{x})=\ln(1+\epsilon )\gtrapprox \ln 1=0} , so just above 0

Lambert W function

≥ e: ln ⁡ x − ln ⁡ ln ⁡ x + ln ⁡ ln ⁡ x 2 ln ⁡ x ≤ W 0 ( x ) ≤ ln ⁡ x − ln ⁡ ln ⁡ x + e e − 1 ln ⁡ ln ⁡ x ln ⁡ x . {\displaystyle \ln x-\ln \ln x+{\frac