Log2

The base-2 (binary) logarithm of z.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageLog2✓↗

Log2(z)the base-2 logarithm of z, $\log_2(z)$.

SageMathlog(z, 2)✓↗SymPylog(z, 2)↗mpmathlog(z, 2)↗Juliaenumeratio_ball(_log2, z)Rustlog2(z)

Typed signatures

Type	Package	Applies
(number) -> number	compute-engine	always

Domain: Elementary functions

Details

• Equivalent to Log(z, 2), and identical to Lb.
• Folds exactly for an integer power of 2 in either direction -- positive ($1024 \to 10$) or negative ($1/8 \to -3$).

Plot

Examples

Basic #

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

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In$\log_{2}(1)$Out$0$

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In$\log_{2}(1\,024)$Out$10$

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

A machine-precision argument evaluates numerically

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In$\log_{2}(1.414\,2)$Out$0.499\,986\,164\,421\,874\,256\,467$

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

Scope #

An exact negative power of 2 folds too: $\log_2 \tfrac18 = -3$

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

A symbolic argument becomes the two-argument Log

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In$\log_{2}(x)$Out$\log_{2}(x)$

An integer power is pulled out: $2\log_2 3$

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

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In$N(\log_{2}(9))$Out$3.169\,925\,001\,442\,312\,362\,91$

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In$\log_{2}(0)$Out$-∞$

As with Ln, the logarithm of complex infinity is $+\infty$

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In$\log_{2}(\tilde∞)$Out$∞$

Element-wise over a matrix, including the exact $\tfrac12$ entry

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In$\log_{2}(\bigl\lbrack\bigl\lbrack2, 1\bigr\rbrack, \bigl\lbrack0, \frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack-∞, -1\bigr\rbrack\bigr\rbrack$

Interval arithmetic: $\log_2[\tfrac13, 2] = [\log_2 \tfrac13, 1]$

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In$\log_{2}(\mathrm{Interval}(\frac{1}{3}, 2))$Out$\mathrm{Interval}(\log_{2}(\frac{1}{3}), 1)$

Uncertainty propagation, to first order: $\log_2(\tfrac15 \pm 0.01) \approx -2.322 \pm 0.072$

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In$\log_{2}(\mathrm{Around}(\frac{1}{5}, 0.01))$Out$\mathrm{Around}(-2.321\,928\,094\,887\,362, 0.072\,134\,752\,044\,448\,18)$

Derivative: $\frac{1}{x\ln 2}$

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In$\frac{\mathrm{d}\log_{2}(x)}{\mathrm{d}x}$Out$\frac{1}{x\ln(2)}$

To 30 significant digits, every one correctly rounded

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In$N(\log_{2}(3), 30)$Out$1.584\,962\,500\,721\,156\,181\,453\,738\,943\,95$

A 100-digit logarithm in base 2

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In$N(\log_{2}(45), 100)$Out$5.491\,853\,096\,329\,674\,710\,777\,797\,317\,385\,023\,193\,384\,460\,208\,409\,542\,732\,966\,261\,704\,898\,131\,232\,197\,342\,340\,894\,700\,693\,195\,546\,639$

A complex argument in base 2

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In$N(\log_{2}(2+i))$Out$1.160\,964\,047\,443\,681+0.668\,902\,106\,225\,488\,1i$

A matrix function with exact eigenvalues 1 and $-1$, since $\log_2\frac12=-1$

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In$\mathrm{FullSimplify}(\mathrm{MatrixFunction}(\mathrm{Log2}, \bigl\lbrack\bigl\lbrack2, 1\bigr\rbrack, \bigl\lbrack0, 1/2\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack1, \frac{4}{3}\bigr\rbrack, \bigl\lbrack0, -1\bigr\rbrack\bigr\rbrack$

$\log_2\infty=\infty$

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In$\log_{2}(∞)$Out$∞$

Applications #

A float argument folds even for a fractional power of 2

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In$\log_{2}(0.5)$Out$-1.0$

Properties #

Same as Log(z, 2)

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

About

Examples	23
Checked against	Wolfram Language 23 agreeSageMath 20 agreeSymPy 19 agree3 differmpmath 16 agreeJulia 13 agree3 errorRust 12 agree2 error
Crosswalk	Wolfram LanguageSageMathSymPympmathJuliaRust

Implementation

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

See also: Log, Log10, Lb