Use $ \log_{10} 2 \equal{} 0.301,\ \log_{10} 3 \equal{} 0.477,\ \log_{10} 7 \equal{} 0.845$ to find the value of $ \log_{10} (10!)$.
Note that you must answer according to the rules:fractional part of $ 0.5$ and higher is rounded up, and everything strictly less than $ 0.5$ is rounded down,
say $ 1.234\longrightarrow 1.23$. Then find the minimum integer value $ n$ such that $ 10! < 2^{n}$.
1. **Calculate $\log_{10}(10!)$ using given logarithm values:**

   We know that:
   \[
   10! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10
   \]
   Therefore:
   \[
   \log_{10}(10!) = \log_{10}(1) + \log_{10}(2) + \log_{10}(3) + \log_{10}(4) + \log_{10}(5) + \log_{10}(6) + \log_{10}(7) + \log_{10}(8) + \log_{10}(9) + \log_{10}(10)
   \]

2. **Simplify using properties of logarithms:**

   Using the properties of logarithms, we can simplify:
   \[
   \log_{10}(4) = \log_{10}(2^2) = 2\log_{10}(2)
   \]
   \[
   \log_{10}(6) = \log_{10}(2 \times 3) = \log_{10}(2) + \log_{10}(3)
   \]
   \[
   \log_{10}(8) = \log_{10}(2^3) = 3\log_{10}(2)
   \]
   \[
   \log_{10}(9) = \log_{10}(3^2) = 2\log_{10}(3)
   \]
   \[
   \log_{10}(10) = \log_{10}(2 \times 5) = \log_{10}(2) + \log_{10}(5)
   \]

3. **Substitute the given values:**

   Given:
   \[
   \log_{10}(2) = 0.301, \quad \log_{10}(3) = 0.477, \quad \log_{10}(7) = 0.845
   \]
   Also, we know:
   \[
   \log_{10}(5) = \log_{10}\left(\frac{10}{2}\right) = \log_{10}(10) - \log_{10}(2) = 1 - 0.301 = 0.699
   \]

4. **Calculate $\log_{10}(10!)$:**

   \[
   \log_{10}(10!) = \log_{10}(1) + \log_{10}(2) + \log_{10}(3) + 2\log_{10}(2) + \log_{10}(5) + (\log_{10}(2) + \log_{10}(3)) + \log_{10}(7) + 3\log_{10}(2) + 2\log_{10}(3) + (\log_{10}(2) + \log_{10}(5))
   \]
   \[
   = \log_{10}(1) + 8\log_{10}(2) + 4\log_{10}(3) + 2\log_{10}(5) + \log_{10}(7)
   \]
   \[
   = 0 + 8 \times 0.301 + 4 \times 0.477 + 2 \times 0.699 + 0.845
   \]
   \[
   = 0 + 2.408 + 1.908 + 1.398 + 0.845
   \]
   \[
   = 6.559
   \]

5. **Find the minimum integer $n$ such that $10! < 2^n$:**

   We need to find $n$ such that:
   \[
   10! < 2^n
   \]
   Taking the logarithm base 2 of both sides:
   \[
   \log_{2}(10!) < n
   \]
   Using the change of base formula:
   \[
   \log_{2}(10!) = \frac{\log_{10}(10!)}{\log_{10}(2)}
   \]
   \[
   \log_{2}(10!) = \frac{6.559}{0.301} \approx 21.7906
   \]
   Therefore:
   \[
   n = \lceil 21.7906 \rceil = 22
   \]

The final answer is $\boxed{22}$