Think about this expression \[\log n!\]

What can you say about it?

Try some values of \(n\).

Choose a base or work generally. What happens if you change the base?

What do you notice?

and

\[\begin{align*} \log_3 6!\, &= \log_3 (1\times2\times3\times4\times5\times6) \\ &= \log_3 1 + \log_3 2 + \log_3 3 + \log_3 4 +\log_3 5 + \log_3 6 \\ &= 4\log_3 2 + 2 + \log_3 5 \end{align*}\]What do you notice about the coefficients?

Do you always get an integer term, whatever base you use?

If you knew \(\log_2 12!\) could you write down what \(\log_4 12!\) is?

What could \(n\) be if \(\log_5 n!\) has integer term \(3\) when expanded as far as possible?

What can you say about \(n\) if \(\log_6 n!\) has integer term \(4\)?

If you know that \(p\) and \(q\) are primes larger than \(7\) and \(a\) is not prime, can you fill in the boxes in the equation below? \[\log \square! = a\log 2 + \square \log 3 + 3 \log 5 +\square \log 7 + \log p +\log q\]

Note that the numbers in the boxes may or may not be the same. Can you make up a similar question?