Log

The logarithm of z in base b, defaulting to base 10.

WikipediaLogarithm↗MathWorldLogarithm↗WikidataQ11197↗Encyclopedia of MathematicsLogarithm_of_a_number↗nLablogarithm↗Britannicascience/logarithm↗FungrimLog↗

Log(z)the base-10 logarithm of z.

SageMathlog(z, 10)!↗SymPylog(z, 10)↗mpmathlog(z, 10)↗Juliaenumeratio_ball(_log10, z)Rustlog10(z)

Log(z, b)the logarithm of z in base b.

SageMathlog(z, b)!↗SymPylog(z, b)↗mpmathlog(z, b)↗Juliaenumeratio_ball(_log_base, z, b)

Log(z, b = 10)The logarithm of z in base b, defaulting to base 10.

SageMathlog(z, b = 10)!↗SymPylog(z, b = 10)↗mpmathlog(z, b = 10)↗Juliaenumeratio_ball(_log_base, z, b = 10)

Typed signatures

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

Domain: Elementary functions

Details

• compute-engine's default base is 10, not $e$: Log(z) means $\log_{10}(z)$, and Ln is the natural logarithm.
• compute-engine's $\mathrm{Log}(z, b)$ takes the value first (value, then base) and defaults to base 10 in the one-argument form; for the natural logarithm use Ln.
• Product rule: $\log_b(xy) = \log_b(x) + \log_b(y)$. Power rule: $\log_b(x^k) = k \log_b(x)$.

Plot

ComplexPlot

real slice

Examples

4 more kept as data, checked against the oracles

Basic #

Default base is 10

live ↗

In$\log_{10}(100)$Out$2$

Same result written with an explicit base

live ↗

In$\log_{10}(100)$Out$2$

Base 2 via the explicit second argument

live ↗

In$\log_{2}(8)$Out$3$

Scope #

$\log_3 1.5$ at machine precision

live ↗

In$\log_{3}(1.5)$Out$0.369\,070\,246\,428\,542\,562\,9$

live ↗

In$\log_{3}(9)$Out$2$

Threads over a list of values

live ↗

In$\log_{2}(\bigl\lbrack2, 4, 8\bigr\rbrack)$Out$\bigl\lbrack1, 2, 3\bigr\rbrack$

Threads over a list of bases

live ↗

In$\log_{\bigl\lbrack2, e, 10\bigr\rbrack}(2.5)$Out$\bigl\lbrack1.321\,928\,094\,887\,362\,347\,87, 0.916\,290\,731\,874\,155\,144\,52, 0.397\,940\,008\,672\,037\,609\,573\bigr\rbrack$

Derivative in base 3: $\frac{1}{x\ln 3}$

live ↗

In$\frac{\mathrm{d}\log_{3}(x)}{\mathrm{d}x}$Out$\frac{1}{x\ln(3)}$

$\log_2\tfrac18 = -3$; an exact negative power of the base folds too

live ↗

In$\log_{2}(\frac{1}{8})$Out$-3$

Properties #

Confirms the default base is 10

live ↗

In$\log_{10}(100)=\log_{10}(100)$Out$\mathrm{True}$

Product rule: $\log_b(xy) = \log_b(x) + \log_b(y)$

live ↗

In$\log_{2}(8×4)=\log_{2}(8)+\log_{2}(4)$Out$\mathrm{True}$

Power rule: $\log_b(x^k) = k \log_b(x)$

live ↗

In$\log_{2}(2^5)=5\log_{2}(2)$Out$\mathrm{True}$

Possible issues #

$\log_{10}(0) = -\infty$

live ↗

In$\log_{10}(0)$Out$-∞$

A negative argument stays symbolic, with the default base filled in; wrap in N(...) for the complex value

live ↗

In$\log_{10}(-5)$Out$\log_{10}(-5)$

About

Examples	18
Checked against	Wolfram Language 15 agree3 differSageMath 15 agree3 differSymPy 14 agree4 differmpmath 14 agree3 differJulia 13 agree3 differ1 errorRust 11 agree6 error
Crosswalk	WikipediaMathWorldWikidataEncyclopedia of MathematicsnLabBritannicaFungrimSageMath ×2SymPy ×2mpmath ×2Julia ×2Rust

Implementation

mappedsympypackages/oracle/src/mappings.ts

compute-engine's one-argument Log is base 10, not the natural logarithm; SymPy's log takes the value first and then the base.

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

The quotient of natural logarithms: std has no logarithm to a base.

See also: Ln, Log2, Log10, Lb