On a blackboard the product $log_{( )}[ ]\times\dots\times log_{( )}[ ]$ is written (there are 50 logarithms in the product). Donald has $100$ cards: $[2], [3],\dots, [51]$ and $(52),\dots,(101)$. He is replacing each $()$  with some card of form $(x)$ and each $[]$ with some card of form $[y]$. Find the difference between largest and smallest values Donald can achieve.
To solve this problem, we need to understand the properties of logarithms and how the product of logarithms can be manipulated. The key insight is that the product of logarithms can be expressed as a ratio of products of logarithms.

1. **Understanding the Product of Logarithms**:
   The product of logarithms can be written as:
   \[
   \prod_{i=1}^{50} \log_{(a_i)}[b_i]
   \]
   where \(a_i\) and \(b_i\) are chosen from the sets \((52), (53), \ldots, (101)\) and \([2], [3], \ldots, [51]\) respectively.

2. **Using the Change of Base Formula**:
   Recall the change of base formula for logarithms:
   \[
   \log_{a}(b) = \frac{\log(b)}{\log(a)}
   \]
   Applying this to our product, we get:
   \[
   \prod_{i=1}^{50} \log_{(a_i)}[b_i] = \prod_{i=1}^{50} \frac{\log(b_i)}{\log(a_i)}
   \]

3. **Simplifying the Product**:
   The product of fractions can be written as a single fraction:
   \[
   \prod_{i=1}^{50} \frac{\log(b_i)}{\log(a_i)} = \frac{\prod_{i=1}^{50} \log(b_i)}{\prod_{i=1}^{50} \log(a_i)}
   \]

4. **Choosing the Cards**:
   Since Donald has to use each card exactly once, the numerator will be the product of the logarithms of the numbers from 2 to 51, and the denominator will be the product of the logarithms of the numbers from 52 to 101. Thus, we have:
   \[
   \frac{\prod_{i=2}^{51} \log(i)}{\prod_{j=52}^{101} \log(j)}
   \]

5. **Calculating the Difference**:
   The largest value of the product occurs when the ratio is maximized, and the smallest value occurs when the ratio is minimized. However, since each card is used exactly once, the product is fixed and does not change regardless of the order in which the cards are used. Therefore, the difference between the largest and smallest values is zero.

Conclusion:
\[
\boxed{0}
\]