Some things to think about

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?

For example, do you agree with these \[\begin{align*} \log_2 3! \, &= \log_2 (1\times 2\times 3) \\ &= \log_2 1 + \log_2 2 + \log_2 3 \\ &= 1+\log_2 3 \end{align*}\]

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?