Simplify a log of a log and series

Question

I'm having a bit trouble trying to write this pattern out

if $$ f_k(x) = x \cdot \log(x) \cdot \log(\log(x)) \cdot \ldots \cdot \log(\ldots(\log(x))). $$ where the final term has k logs in it (So $f_0(x) = x$, $f_1(x) = x \log(x)$, $f_2(x) = x \log(x) \log(\log(x))$, and so on). then how would I write the final term?

Is there a way to simplify writing a log of a log?

Answer 1

Welcome to MSE!

If you want to indicate you're composing a function $n$ times, it's standard to write

$$f^{\circ n}(x).$$

That is

• $f^{\circ 0}(x) = x$
• $f^{\circ 1}(x) = f(x)$
• $f^{\circ 2}(x) = f(f(x))$
• $f^{\circ 3}(x) = f(f(f(x)))$
• etc.

So your term is the product of $\log^{\circ j}$ for $j \leq k$. Or, in symbols:

$$ f_k(x) = \prod_{j = 0}^k \log^{\circ j}(x) $$

If you aren't familiar with $\prod$ as a product, you can see more here.

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I hope this helps ^_^