Does hitting time have scaling property or self-similar proerty

Question

Consider the hitting time such that $\tau_x = \inf\{t>0: B_t = x\}$ where $B_t$ is a standard brownian motion. Can $\tau_x$ be scaled like that $\tau_{ax}=a^2\tau_x$ for $\forall a>0, a\in \mathbb R $? I have checked the probability density function of $\tau_x$ satisfy the scaling property since $f_{\tau_{x}}(t)=\frac{|x|e^{-\frac{x^2}{2t}}} { \sqrt{2\pi}t^{\frac{3}{2}}}$ and it can be proved that $f_{\tau_{ax}}(t)=\frac{|ax|e^{-\frac{(ax)^2}{2t}}} { \sqrt{2\pi}t^{\frac{3}{2}}}=a^{-3}\frac{a|x|e^{-\frac{(x)^2}{2t/a^2}}} { \sqrt{2\pi}(\frac{t}{a^2})^{\frac{3}{2}}}=a^{-2}f_{\tau_{x}}(\frac{t}{a^2})$. Did I make any mistakes on these?