I was just doing some time-pass with my calculator but then I observe something.I don't know whether it is senseful to ask.So here's my question. ln ln (1) is not defined but for all values greater than 1 it is defined.So then I try to find values for which ln ln ln(x) is defined,then I get to know that it get's defined from 2.72.If ln is taken 4 times it's start giving values from 15.2.So my question is if ln is given particular times how I can come to know the infimum of values for which it is defined?
Answer
$\ln (x)$ is defined for $x>0$
$\ln (\color{blue}{\ln (x)})$ will be defined for $\color{blue}{\ln (x)}>0 \implies x >1$
$\ln (\color{blue}{\ln ( \ln (x))})$ is defined for $\color{blue}{\ln ( \ln (x))}>0 \implies \color{blue}{ \ln (x)}>1 \implies x>e$
You see the pattern now?
$$0, e^0, e^1, e^e, e^{e^e} \ldots$$
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