In 2007, Andrews and Paule introduced the broken $k$-diamond partition function $Δ_k(n)$, which has received a lot of researches on the arithmetic propertises. In this paper, we prove that $D^3\log Δ_1(n-1)>0$ for $n\geq 5$ and $D^3 \log Δ_2(n-1)>0$ for $n\geq 7$, where $D$ is the difference operator with respect to $n$. We also conjecture that for any $k\geq 1$ and $r\geq 1$, there exists a positive integer $n_k(r)$ such that for $n\geq n_{k}(r)$, $(-1)^r D^r \log Δ_k(n)>0$. This is analogous to the positivity of finite differences of the logarithm of the partition function, which has been proved by Chen, Wang and Xie. Furthermore, we obtain that both $\{Δ_1(n)\}_{n\geq 0}$ and $\{Δ_2(n)\}_{n\geq 0}$ satisfy the higher order Turán inequalities for $n \geq 6$.