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PMO National Stage Oral Phase

Algebra Difficulty 4.4 AIME Prove it Philippines

Problem:
Let $f(x) = \ln x$ . What are the values of $x$ in the domain of $(f \circ f \circ f \circ f \circ f)(x)$ ?

Solution

Solution:
We have $(f \circ f \circ f \circ f \circ f)(x) = \ln (\ln (\ln (\ln (\ln (x)))))$ .

The domain of this function must satisfy $\ln (\ln (\ln (\ln (x)))) > 0$ , which implies that $\ln (\ln (\ln (x))) > 1$ , $\ln (\ln (x)) > e$ , $\ln (x) > e^{e}$ , and $x > e^{e^{e}}$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.