The digit in the unit's place of the product $81\times 82\times \cdots \times 89$ is:

The given series is a product of consecutive integers from $81$ to $89$:
$$\text{Product}=81\times 82\times 83\times 84\times 85\times 86\times 87\times 88\times 89$$

Notice that the series contains the numbers $82$ and $85$.

* The unit digit of $82$ is $2$.
* The unit digit of $85$ is $5$.

When we multiply these two unit digits together:

$$2\times 5=10$$

Any number multiplied by $10$ or any product that contains a combination of $2$ and $5$ will inevitably have $0$ in its unit's place.
Therefore, the unit's digit of the entire product is $0$.