What is a logarithm in the light of group theory?

Question

Logarithms connect the operation of addition and the operation of multiplication.

How does group theory sheds light to this property of logarithms?

Answer 1

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 $\Bbb R$.

Answer 2

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.