How many pairs of ${b_i}^k \equiv g_i\pmod p$ are enough to solve discrete logarithm problem?
Having multiple sets $b_i^k = g_i$ with a common solution $k$ do not help you recover $k$; it cannot make the problem much easier than if you had the simple equation $b^k = g$.

Here's the proof; suppose that there was an efficient way to recover $k$ given a $n$ different $b_i^k = g_i$ equations.

Then, we can efficiently solve the simple equation $b^k = g$ by doing the following:


We select $n$ random values $r_1, ..., r_n$
We compute:


$$b_i = b^{r_i}$$

$$g_i = g^{r_i}$$


We have our $n$ equations, as $b_i^k = (b^{r_i})^k = (b^k)^{r_i} = g^{r_i} = g_i$, so we use the efficient way to recover $k$ given $n$ equations.


We then know the solution $k$ to the original equation, and the only extra time we took was to the time select $n$ random values and to compute $2n$ modular exponentiations.