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Can there be a numerical system in which logarithms can be expressed in terms of exponentials in closed form?
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Can there be a numerical system in which logarithms can be expressed in terms of exponentials in closed form?
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Can there be a numerical system in which logarithms can be expressed in terms of exponentials in closed form?
@Ben McKay fine, I would not expect it to be an algebraic extension. By "numerical field" I understood a field that would have the majority of properties of real/complex numbers (that is commutativity and associativity of multiplication etc). At least what allows to call say, hyperreal numbers still "numbers". 
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Can there be a numerical system in which logarithms can be expressed in terms of exponentials in closed form?
@Ben McKay what you are saying is interesting. Will this extension satisfy the usual notions of a "numerical " field? Is it possible to somehow derive other algebraic properties of such extension? 
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asked  Can there be a numerical system in which logarithms can be expressed in terms of exponentials in closed form? 
Apr
24 
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Nonstandard numbers and exponential form of Zeta function
@მამუკა ჯიბლაძე moreover, it can be deduced that $$\sum_{n=0}^\infty \frac1{n+1}=\ln \omega_+$$ which explains why Harmonic series has EulerMasceroni constant as its Ramanujan's sum. 
Apr
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Classifying countable sets of weighted dots on a real line
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Apr
11 
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Classifying countable sets of weighted dots on a real line
@Vidit Nanda I am thinking about a numerical system that would extend the real numbers, each class would correspond to an extended number. There would be rules on arithmetical operations, for instance, multiplication goes as follows: put two real axes perpendicular to each other with sets of the classes you intend to multiply and draw the lines parallel to the axes over all the dots. Find the intersection dots on the plane and put them on a new real axis in order according to the greatest coordinate, multiplying weights in process. The class of this new set will be the product. 
Apr
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Classifying countable sets of weighted dots on a real line
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Apr
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revised 
Classifying countable sets of weighted dots on a real line
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Apr
11 
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Classifying countable sets of weighted dots on a real line
@Vidit Nanda you can move all the negative dots to the symmetric positive positions (where they would coincide with existing dots, you can sum up the weights), you can move the dot from zero (and other finite amount of dots ) freely wherever you want it. 
Apr
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Classifying countable sets of weighted dots on a real line
@Yaakov Baruch I have added a rule for accumulation points, second from the end. 
Apr
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Classifying countable sets of weighted dots on a real line
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Apr
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Classifying countable sets of weighted dots on a real line
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Apr
11 
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Classifying countable sets of weighted dots on a real line
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Apr
11 
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Classifying countable sets of weighted dots on a real line
@Yaakov Baruch what about cases without accumulation point? 
Apr
11 
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Classifying countable sets of weighted dots on a real line
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Apr
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Classifying countable sets of weighted dots on a real line
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Apr
11 
asked  Classifying countable sets of weighted dots on a real line 
Feb
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awarded  Necromancer 