Equational identities of real multiplication augmented by a real number

Question

Consider the structure $(\mathbb R, *, r)$, where $r$ is a real number that is neither $0$, $1$, or $-1$. Are the commutative and associative identities already sufficient to derive all the universally valid equations in that structure? I apologize if this question is similar to one I asked previously.

Answer 1

The answer is yes. Suppose an equation $t(x_1,\dots,x_n) = s(x_1,\dots,x_n)$ holds in $\mathbb{R}$, where $t$ and $s$ are terms. Modulo commutativity and associativity, this equation is equivalent to one of the form $$r^{d_0}\prod_{i = 1}^n x_i^{d_i} = r^{e_0}\prod_{i = 1}^n x_i^{e_i},$$ where the $d_i$ and $e_i$ are natural numbers.

Let $K = \mathbb{Q}(r)$ be the smallest subfield of $\mathbb{R}$ containing $r$, and let $a_1,\dots,a_n\in \mathbb{R}$ be algebraically independent over $K$ (we can do this since $K$ is countable). Then $a_1,\dots,a_n$ satisfy the equation above. But viewing this equation as a polynomial equation over $K$ in the variables $x_i$, it must be trivial, i.e. the two polynomials must be equal. Then we have $d_i = e_i$ for all $i>0$, and $r^{d_0} = r^{e_0}$. But since $r\neq -1,0,1$, we must have $d_0 = e_0$ as well.