Question 154
Find all binary operations $\diamond$: $\mathbb{R}_{\gt 0} \times \mathbb{R}_{\gt 0} \to \mathbb{R}_{\gt 0}$ (meaning $\diamond$ takes pairs of positive real numbers to positive real numbers) such that for any real numbers $a, b, c \gt 0$,
* the equation $a \diamond (b \diamond c) = (a \diamond b) \cdot c$ holds; and
* if $a \ge 1$ then $a \diamond a \ge 1$.
* the equation $a \diamond (b \diamond c) = (a \diamond b) \cdot c$ holds; and
* if $a \ge 1$ then $a \diamond a \ge 1$.