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In mathematics, in the field of tropical analysis, the log-semiring is the semiring structure on the logarithmic scale, obtained by considering the extended real numbers as logarithms. That is, the operations of addition and multiplication are defined by conjugation: exponentiate the real numbers, obtaining a positive number, add or multiply these numbers with the ordinary algebraic operations on real numbers, and then take the logarithm 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 ×. These operations depend on the choice of base b for the exponent and logarithm, which corresponds to a scale factor, and are well-defined for any positive base other than 1; using a base b < 1 is equivalent to using a negative sign and using the inverse 1/b > 1. If not qualified, the base is conventionally taken to be e or 1/e, which corresponds to e with a negative.
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