Estimating Difference of Two Averages of Function by Its BMO Norm

Question

Assume $f\in \text{BMO}(\mathbb{R}^d)$. Denote $f_Q=\displaystyle\dfrac{1}{|Q|}\int_Q f(x)\,dx$. I want to show for $\alpha>2$ and any cube $Q$ with positive volume, we have $|f_{\alpha Q}-f_Q|\leq C_d \log(\alpha) \|f\|_{\star}$ where $\alpha Q$ is dilation of $Q$ by factor $\alpha$ and $\|f\|_{\star}$ is BMO norm of $f$. So I know \begin{align*} \sup_Q\inf_{a\in \mathbb{C}} \dfrac{1}{|Q|}\int_Q|f(x)-a|\,dx\leq \|f\|_{\star}. \end{align*} For every $a\in \mathbb{C}$, we have \begin{align*} |f_{\alpha Q}-f_Q|&\leq |f_{\alpha Q}-a|+|f_Q-a|\\ &=\left| \dfrac{1}{|\alpha Q|}\int_{\alpha Q} (f(x)-a)\,dx \right|+\left| \dfrac{1}{| Q|}\int_{ Q} (f(x)-a)\,dx \right|\\ &\leq \dfrac{1}{|\alpha Q|}\int_{\alpha Q} |f(x)-a|\,dx+\dfrac{1}{| Q|}\int_{ Q} |f(x)-a|\,dx. \end{align*} Now take infimum over all $a\in \mathbb{C}$ from both sides, and taking supremum over all cubes on the right hand side, we got the right hand-side is at most $2\|f\|_{\star}$. What is wrong with this approach since I even got the constant does not depend on $d$ and $\alpha$?