Four enthusiastic mathematicians are asked to think of a function involving the number 100. The challenge is to think of the function which is biggest for big values of n

Archimedes chooses a logarithm function

$A (n) = \log (100 n)$
Bernoulli decides to take 100th powers

$B (n) = n^{100}$
Copernicus takes powers of 100

$C (n) = 100^{n}$
and, finally, de Moivre, who likes to be different, chooses the factorial function which he claims will be quite big enough without any reference to 100 at all

$D (n) = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1$

Which function is biggest for large values of n? Can you determine a value beyond which you know this function will be biggest?

[Extension: To find the exact switch-over value will be difficult and will require the clever use of a spreadsheet or computer.]

What could you say if the 100s were replaced by a million? billions? Create a convincing argument to prove your results to the mathematicians.