Fluency exercise

## 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?