Logarithms connect the operation of addition and the operation of multiplication.
How does group theory sheds light to this property of logarithms?
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See also https://math.stackexchange.com/questions/3061985/what-makes-the-pairs-of-operators-and-÷-×-so-similar – lhf May 04 '19 at 10:58
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What this tells us is $(\Bbb R,\,+)$ is isomorphic to $(\Bbb R^+,\,\times)$. (I'm using $\Bbb R^+$ as a symbol for $(0,\,\infty)$ rather than as ring-theoretic notation.) When groups are isomorphic, a specific function called an isomorphism transforms one into the other while preserving the group structure, in this case the function $\ln x$ from $\Bbb R^+$ to $\Bbb R$.
J.G.
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7This is indeed an answer; it's probably not a good answer from the perspective of the original asker, but it's a quite good answer for those with a bit more experience. – John Hughes May 04 '19 at 11:02
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1For a ring $R$, I've always seen the notation $R^+$ used to denote the underlying additive group of $R$, so I was a bit confused by the expression $(\Bbb R^+,,\times)$. – Servaes May 04 '19 at 11:18
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Let $G$ be a group with binary operation $\ast$.
For $g\in G$, define $g^1 = g, \,g^2 = g \ast g, \,g^3 = g \ast g \ast g, \,\cdots$
If $a,\,b\in G$ and there is an integer $k$ such that $b^k = a$, then we say that $k$ is the discrete logarithm of $a$ base $b$, and write $k = \log_b a$.
Finding discrete logarithms may be very hard, which makes them useful in cryptography.
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1+1 this is an interesting interpretation of the OP's question, although it's probably not what the OP means. (You should use mathjax formatting on this site: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference .) – Ethan Bolker May 04 '19 at 15:43
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I've edited your answer to use MathJax as per @EthanBolker's suggestion. This included using
\astfor the binary operation. If you'd prefer to edit your answer to use $\times$ instead, just use\timesinstead. – J.G. May 05 '19 at 06:33