You might like to think about whether you agree with these
$$\begin{aligned}
\log_2 3! \, &= \log_2 (1\times 2\times 3) \\
&= \log_2 1 + \log_2 2 + \log_2 3 \\
&= 1+\log_2 3
\end{aligned}$$
- 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?
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