Logarithm

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This page describes the dyadic arithmetic function. For the monadic natural logarithm function, see Natural Logarithm.
⍟

Logarithm (⍟), or Log, is a dyadic scalar function which computes the logarithm of the two arguments. More precisely, <syntaxhighlight lang=apl inline>X⍟Y</source> computes how much power of X equals Y, i.e. the value of R that satisfies <syntaxhighlight lang=apl inline>Y=X*R</source>. Logarithm shares the glyph <syntaxhighlight lang=apl inline>⍟</source> with the monadic arithmetic function Natural Logarithm. The glyph, a composition of the glyphs for Circular (<syntaxhighlight lang=apl inline>○</source>) and Power (<syntaxhighlight lang=apl inline>*</source>) to indicate its close mathematical ties with these two functions, is a stylised tree log.[1]

Examples

<syntaxhighlight lang=apl> 

     2⍟0.5 1 2 32 1024

¯1 0 1 5 10 </source>

Logarithm can be used to determine how many digits are needed to write a positive integer Y in base X:

<syntaxhighlight lang=apl> 

     Digits←{1+⌊⍺⍟⍵}
     ToBase←⊥⍣¯1
     (2 Digits 100) (2 ToBase 100)

┌─┬─────────────┐ │7│1 1 0 0 1 0 0│ └─┴─────────────┘ 

     (10 Digits 100) (10 ToBase 100)

┌─┬─────┐ │3│1 0 0│ └─┴─────┘

</source>

Works in: Dyalog APL

Properties

By definition, logarithm is the inverse of the power with the same base (left argument).

<syntaxhighlight lang=apl> 

     2*1 2 3 4 5

2 4 8 16 32 

     2⍟2 4 8 16 32

1 2 3 4 5 

     2 (*⍣¯1 ≡ ⍟) ⍳10

1

</source>

Works in: Dyalog APL

Reciprocal on the left or right argument gives the negated result.

<syntaxhighlight lang=apl> 

     2⍟÷2 4 8 16 32

¯1 ¯2 ¯3 ¯4 ¯5 

     (÷2)⍟2 4 8 16 32

¯1 ¯2 ¯3 ¯4 ¯5 </source>

See also

External links

Documentation

References

  1. ↑ McDonnell, E. E.. Recreational APL: The Story of <syntaxhighlight lang=apl inline>○</source>. APL Quote-Quad, Volume 8, Number 2, 1977-12.
APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
Monadic Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root
Dyadic Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
Structural Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
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Operators Monadic Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis
Dyadic Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator)
Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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