Prove that $r$-ary entropy $H_r(s) = H(s) / \log_2 r$
There's nothing to prove here, this follows by Definition. Assume a discrete r.v. $X$, then

$$H(X) \triangleq \mathbb{E}[-\log \Pr(X)]$$

and,

$$H_r (X) \triangleq \mathbb{E}[-\log_r \Pr(X)].$$

Therefore, the claim follows (using $E(aX)=aE(X)$ for a constant $a$). Note that in your question you use $H(X)$ as the binary entropy $H(X) \triangleq H_2(X) = \mathbb{E}[-\log_2 \Pr(X)]$, hence the multiplicative factor is $\log_2 r$.