Inmathematics, in the field oftropical analysis, thelog semiring is thesemiring structure on thelogarithmic scale, obtained by considering theextended real numbers aslogarithms. That is, the operations of addition and multiplication are defined byconjugation:exponentiate the real numbers, obtaining a positive (or zero) number, add or multiply these numbers with the ordinaryalgebraic operations on real numbers, and then take thelogarithm to reverse the initial exponentiation. Such operations are also known as, e.g.,logarithmic addition, etc. As usual in tropical analysis, the operations are denoted by ⊕ and ⊗ to distinguish them from the usual addition + and multiplication × (or ⋅). These operations depend on the choice of baseb for the exponent and logarithm (b is a choice oflogarithmic unit), which corresponds to a scale factor, and are well-defined for any positive base other than 1; using a baseb < 1 is equivalent to using a negative sign and using the inverse1/b > 1.^[a] If not qualified, the base is conventionally taken to bee or1/e, which corresponds toe with a negative.

The log semiring has thetropical semiring as limit ("tropicalization", "dequantization") as the base goes to infinity⁠${\displaystyle b\to \mathrm{\infty }}$⁠ (max-plus semiring) or to zero⁠${\displaystyle b\to 0}$⁠ (min-plus semiring), and thus can be viewed as adeformation ("quantization") of the tropical semiring. Notably, the addition operation,logadd (for multiple terms,LogSumExp) can be viewed as a deformation ofmaximum orminimum. The log semiring has applications inmathematical optimization, since it replaces the non-smooth maximum and minimum by a smooth operation. The log semiring also arises when working with numbers that are logarithms (measured on alogarithmic scale), such asdecibels (seeDecibel § Addition),log probability, orlog-likelihoods.

The operations on the log semiring can be defined extrinsically by mapping them to the non-negative real numbers, doing the operations there, and mapping them back. The non-negative real numbers with the usual operations of addition and multiplication form asemiring (there are no negatives), known as theprobability semiring, so the log semiring operations can be viewed aspullbacks of the operations on the probability semiring, and these areisomorphic as rings.

Formally, given the extended real numbersR ∪ {–∞, +∞}^[b] and a baseb ≠ 1, one defines:

Regardless of base, log multiplication is the same as usual addition,${\displaystyle x{\otimes }_{b}y=x+y}$, since logarithms take multiplication to addition; however, log addition depends on base. The units for usual addition and multiplication are 0 and 1; accordingly, the unit for log addition is${\displaystyle {\log }_{b}0=-\mathrm{\infty }}$ for${\displaystyle b>1}$ and${\displaystyle {\log }_{b}0=-{\log }_{1/b}0=+\mathrm{\infty }}$ for, and the unit for log multiplication is${\displaystyle \log 1=0}$, regardless of base.

More concisely, the unit log semiring can be defined for basee as:

with additive unit−∞ and multiplicative unit 0; this corresponds to the max convention.

The opposite convention is also common, and corresponds to the base1/e, the minimum convention:^[1]

with additive unit+∞ and multiplicative unit 0.

A log semiring is in fact asemifield, since all numbers other than the additive unit−∞ (or+∞) has a multiplicative inverse, given by${\displaystyle -x,}$ since${\displaystyle x\otimes -x={\log }_b(b^x\cdot b^{-x})={\log }_b(1)=0.}$ Thus log division ⊘ is well-defined, though log subtraction ⊖ is not always defined.

A mean can be defined by log addition and log division (as thequasi-arithmetic mean corresponding to the exponent), as

${\displaystyle M_{\mathrm{l}\mathrm{m}}(x,y):=(x\oplus y)\oslash 2={\log }_b((b^x+b^y)/2)={\log }_b(b^x+b^y)-{\log }_{b}2=(x\oplus y)-{\log }_{b}2.}$

This is just addition shifted by${\displaystyle -{\log }_{b}2,}$ since logarithmic division corresponds to linear subtraction.

A log semiring has the usual Euclidean metric, which corresponds to thelogarithmic scale on thepositive real numbers.

Similarly, a log semiring has the usualLebesgue measure, which is aninvariant measure with respect to log multiplication (usual addition, geometrically translation) with corresponds to thelogarithmic measure on theprobability semiring.

• Logarithmic mean
• LogSumExp
• Softmax

• Logarithms
• Tropical analysis

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