Find the number of solutions for $\underbrace{x*x*\ldots*x}_{x\text{ 10 times}}=\frac{1}{10}$ with a given binary operation.

Solution 1:

What a strange operation! The trick is to find an isomorphism, and they have given you the hint: because $f$ is a bijection, we can say that the operation of $*$ on $(-1, 1)$ is isomorphic to the operation of usual multiplication on $(0, \infty)$. Thus, in terms of $y = f(x) \in (0, +\infty)$, the problem is simply to solve $y^{10} = f(1/10) \in (0, +\infty)$, which has exactly one solution.

(One should check, quite painlessly, that having an isomorphism is "really this good"; for example, that we also get $x * ... * x = f^{-1}(f(x)^n)$ and so on. Indeed, a bijection satisfying the homomorphism law is the correct notion of an isomorphism between magmas.)