Is the Discrete logarithm problem suitable for this pairing scheme? Let $Ans$ be the product of two pairings : $e(g,h)^{k} \times e(g,h)^{r}=Ans$

If everybody knows only $[g,h,e(g,h)^{k}]$ but $[r,Ans]$ is not known. In the discrete logarithm problem, the user knows two values. In this scheme, the user only know one value.

Is the discrete logarithm problem suitable for this scheme or not?
First of all, let us simplify the equation by replacing things that the attacker can compute with known constants.  We come up with:

$$a \cdot b^x = y$$

where the attacker knows $a$ (which is $e(g,h)^k$) and $b$ (which is $e(g, h)$, which he can compute, as he knows $g, h$), and the attacker solves for $x, y$.

If it is sufficient for an attacker to find a solution, he can do that easily; he can select a random $x$, and solve it for $y$; there's a solution.

If he needs to find "the correct" solution (that is, the one possible $x, y$ pair that satisfies additional criteria), he doesn't have enough information to pick out which one it is.  He might be able to if he knows what that additional criteria is; it would depend on what that looks like.