Ln

The natural logarithm, the inverse of Exp.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaNatural logarithm↗MathWorldNaturalLogarithm↗WikidataQ204037↗Britannicatopic/natural-logarithm↗DLMF4.2.E2↗FungrimLog!↗Fungrim entryWolfram LanguageLog✓↗

Ln(x)the natural logarithm of x, base $e$.

SageMathlog(x)✓↗SymPylog(x)↗mpmathlog(x)↗Juliaenumeratio_ball(log, x)Rustln(x)

Typed signatures

Type	Package	Applies
(complex | infinity, base: (complex | infinity)?) -> complex | infinity	analyticreplaces compute-engine	always
(complex | infinity, base: (complex | infinity)?) -> complex | infinity	compute-engine	always

Domain: Elementary functions

Details

• Inverse of Exp: $\ln(e^x) = x$.
• The natural logarithm. Note compute-engine's own Log is not a synonym: it defaults to base 10 (see that entry).
• $\ln(0) = -\infty$. A negative real argument reduces to the principal value $\ln(-q) = \ln(q) + i\pi$ for a positive rational $q$ (@enumeratio/analytic); compute-engine's own plain evaluation would otherwise leave it symbolic.

Plot

ComplexPlot

real slice

Examples

5 more kept as data, checked against the oracles

Basic #

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In$\ln(1)$Out$0$

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In$\ln(e)$Out$1$

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In$\ln(0)$Out$-∞$

A machine-precision argument evaluates numerically

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In$\ln(1.5)$Out$0.405\,465\,108\,108\,164\,381\,978$

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In$\operatorname{Series}(\ln(x+1), x, 0, 10)$Out$x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}-\frac{x^6}{6}+\frac{x^7}{7}-\frac{x^8}{8}+\frac{x^9}{9}-\frac{x^{10}}{10}+O\left(x^{11}\right)$

Scope #

$\mathrm{Ln}(-1) = i\pi$, the principal value past the branch cut, overriding compute-engine's plain policy of leaving a negative-real Ln symbolic

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In$\ln(-1)$Out$i π$

N(...) for a decimal value

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In$N(\ln(1\,000))$Out$6.907\,755\,278\,982\,137\,052\,05$

Arbitrary precision via N(...)

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In$N(\ln(\frac{5}{2}))$Out$0.916\,290\,731\,874\,155\,065\,184$

Evaluates at a complex argument

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In$\ln(2.5+i)$Out$0.990\,500\,734\,433\,291\,704\,176+0.380\,506\,377\,112\,364\,9i$

A negative float gives the principal complex value

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In$\ln(-2.5)$Out$0.916\,290\,731\,874\,155\,065\,184+3.141\,592\,653\,589\,793i$

Threads element-wise over a list

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In$\ln(\bigl\lbrack1, 2.1, 3.1, 4.1\bigr\rbrack)$Out$\bigl\lbrack0, 0.741\,937\,344\,729\,377\,312\,483, 1.131\,402\,111\,491\,100\,561\,91, 1.410\,986\,973\,710\,262\,119\,85\bigr\rbrack$

An integer power is pulled out: $\ln 1000 = 3\ln 10$

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In$\ln(1\,000)$Out$3\ln(10)$

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In$\ln(e^3)$Out$3$

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In$\ln(∞)$Out$∞$

$|\ln z| \to \infty$ and its real part goes to $+\infty$: $\ln(\tilde\infty) = +\infty$

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In$\ln(\tilde∞)$Out$∞$

$\ln i = i\pi/2$ exactly

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In$\ln(i)$Out$\frac{1}{2}i π$

A unit fraction becomes a negated logarithm: $\ln\tfrac12 = -\ln 2$

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In$\ln(\frac{1}{2})$Out$-\ln(2)$

Interval arithmetic: $\ln$ is increasing, so $\ln[\tfrac13, e]$ is the interval of the endpoint values -- $\ln(1/3)$ folds to $-\ln 3$ (see the unit-fraction example above)

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In$\ln(\mathrm{Interval}(\frac{1}{3}, e))$Out$\mathrm{Interval}(-\ln(3), 1)$

Uncertainty propagation: $\ln(2 \pm 0.01) \approx 0.693 \pm 0.005$

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In$\ln(\mathrm{Around}(2, 0.01))$Out$\mathrm{Around}(0.693\,147\,180\,559\,945\,3, 0.005)$

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In$\ln(e^{x})$Out$x$

Derivative: $1/x$

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In$\frac{\mathrm{d}\ln(x)}{\mathrm{d}x}$Out$\frac{1}{x}$

Antiderivative: $x\ln x - x$

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In$\int\!\ln(x)\, \mathrm{d}x$Out$x\ln(x)-x$

Mercator series for $\ln(1 + x)$

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In$\operatorname{Series}(\ln(1+x), x, 0, 4)$Out$x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+O\left(x^5\right)$

To 40 significant digits, every one correctly rounded

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In$N(\ln(2), 40)$Out$0.693\,147\,180\,559\,945\,309\,417\,232\,121\,458\,176\,568\,075\,5$

$\ln(-\tfrac12) = i\pi - \ln 2$ — a negative rational, folded through the unit-fraction rule too.

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In$\ln(-\frac{1}{2})$Out$i π-\ln(2)$

N(...) decimalizes $\ln(i)$, not just the exact $i\pi/2$ form

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In$N(\ln(i))$Out$1.570\,796\,326\,794\,896\,6i$

Ln(-Infinity) is +Infinity: $\ln(-x) = \ln(x) + i\pi$, and the magnitude alone diverges as $x \to +\infty$

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In$\ln(-∞)$Out$∞$

Known value · DLMF 4.2.5

A 50-digit value of $\ln\frac52$

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In$N(\ln(\frac{5}{2}), 50)$Out$0.916\,290\,731\,874\,155\,065\,183\,527\,211\,768\,011\,071\,450\,101\,219\,908\,26$

The domain is the positive reals

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In$\mathrm{FunctionDomain}(\ln(x), x)$Out$0\lt x$

The range is all of the reals

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In$\mathrm{FunctionRange}(\ln(x), x, y)$Out$\mathrm{True}$

The logarithm is not analytic on the whole real line

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In$\mathrm{FunctionAnalytic}(\ln(x), x)$Out$\mathrm{False}$

Not meromorphic either

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In$\mathrm{FunctionMeromorphic}(\ln(x), x)$Out$\mathrm{False}$

Strictly increasing, so one-to-one

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In$\mathrm{FunctionInjective}(\ln(x), x)$Out$\mathrm{True}$

Onto the real line

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In$\mathrm{FunctionSurjective}(\ln(x), x)$Out$\mathrm{True}$

Derivative $\frac1z$

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In$\frac{\mathrm{d}\ln(z)}{\mathrm{d}z}$Out$\frac{1}{z}$

Chain rule: $\frac{d}{dx}\ln\ln x = \frac{1}{x\ln x}$

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In$\frac{\mathrm{d}\ln(\ln(x))}{\mathrm{d}x}$Out$\frac{1}{x\ln(x)}$

Ln undoes the exponential: $\ln e^n = n$

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In$\ln(e^{n})$Out$n$

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In$\operatorname{Series}(\ln(x), x, 1, 4)$Out$-1+x-\frac{(x-1)^2}{2}+\frac{(x-1)^3}{3}-\frac{(x-1)^4}{4}+O\left((x-1)^5\right)$

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In$\operatorname{Series}(\ln(x), x, -2, 3)$Out$i π+\ln(2)-\frac{x+2}{2}-\frac{(x+2)^2}{8}-\frac{(x+2)^3}{24}+O\left((x+2)^4\right)$

Properties #

Inverse of Exp: $\ln(e^x) = x$

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In$\ln(e^5)=5$Out$\mathrm{True}$

Product rule: $\ln(ab) = \ln a + \ln b$

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In$\ln(2)+\ln(3)=\ln(6)$Out$\mathrm{True}$

PowerExpand splits a logarithm of a product, assuming positive factors

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In$\mathrm{PowerExpand}(\ln(xy))$Out$\ln(x)+\ln(y)$

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In$N(\ln(2)^2)$Out$0.480\,453\,013\,918\,201\,424\,667$

Known value · Wolfram MathematicalFunctionData, Log:Natural, Bailey-Borwein-Plouffe formula

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In$N(\ln(2)^3)$Out$0.333\,024\,651\,988\,929\,479\,719$

Known value · Wolfram MathematicalFunctionData, Log:Natural, Bailey-Borwein-Plouffe formula

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In$N(\ln(2)^4)$Out$0.230\,835\,098\,583\,083\,451\,887$

Known value · Wolfram MathematicalFunctionData, Log:Natural, Bailey-Borwein-Plouffe formula

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In$N(\ln(2)^5)$Out$0.160\,002\,697\,757\,141\,321\,081$

Known value · Wolfram MathematicalFunctionData, Log:Natural, Bailey-Borwein-Plouffe formula

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In$N(\ln(2)^2)$Out$0.480\,453\,013\,918\,201\,424\,667$

Known value · Wolfram MathematicalFunctionData, Log:Natural, Bailey-Borwein-Plouffe formula

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In$N(\ln(2)^3)$Out$0.333\,024\,651\,988\,929\,479\,719$

Known value · Wolfram MathematicalFunctionData, Log:Natural, Bailey-Borwein-Plouffe formula

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In$N(\ln(2)^4)$Out$0.230\,835\,098\,583\,083\,451\,887$

Known value · Wolfram MathematicalFunctionData, Log:Natural, Bailey-Borwein-Plouffe formula

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In$N(\ln(2)^5)$Out$0.160\,002\,697\,757\,141\,321\,081$

Known value · Wolfram MathematicalFunctionData, Log:Natural, Bailey-Borwein-Plouffe formula

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In$N(\int_{1}^{2}\!\frac{\ln(t)^2}{t-1}\, \mathrm{d}t)$Out$0.30051422578989856±1.1233763348977733e-20$

Known value · Wolfram MathematicalFunctionData, Log:Natural, Nielsen‐Ramanujan constants

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In$N(\int_{0}^{∞}\!\frac{\ln(t)^2}{e^{t}}\, \mathrm{d}t)$Out$1.978111990655945±4.2166270475263445e-13$

Known value · Wolfram MathematicalFunctionData, Log:Natural, Euler-Mascheroni integral

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In$N(\int_{0}^{1}\int_{0}^{1}\!\frac{\ln(xy)}{1-xy}\, \mathrm{d}x \mathrm{d}y)$Out$-2.4041138063191885±6.4377781180246115e-12$

Known value · Wolfram MathematicalFunctionData, Log:Natural, Beukers's formula

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In$N(\int_{1}^{2}\!\frac{\ln(t)^3}{t-1}\, \mathrm{d}t)$Out$0.14251419793571093±1.4339268116254447e-21$

Known value · Wolfram MathematicalFunctionData, Log:Natural, Nielsen‐Ramanujan constants

About

Examples	59
Checked against	Wolfram Language 59 agreeSageMath 48 agree2 errorSymPy 46 agree2 differ3 unevaluatedmpmath 33 agree1 differJulia 32 agree3 errorRust 21 agree9 error
Crosswalk	WikipediaMathWorldWikidataBritannicaDLMFFungrimFungrim entry ×93Wolfram LanguageSageMathSymPympmathJuliaRust

Implementation

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

See also: Exp, Log, Log2