Domain of $\ln(\ln(\ln(x)))$?

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So my problem states,

Find the Domain and Derivative of the Function $$\ln(\ln(\ln(x))).$$

I know that I can just use the chain rule to find the derivative, but to find the domain, I don't know. All I know is that I need to use the identity $$e^{\ln x}=x.$$

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2 Answers

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You need $\ln \ln x$ to be positive for $\ln \ln \ln x$ to be real, which means $\ln x $ has to be greater than $1$ , which means that $x > e$.

As for derivatives, use the chain rule and $\frac{d}{dx} \ln x = \frac{1}{x}$.

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1. domain - ln(ln x)) > 0 => ln x > 1 => x > e

2. derivative - chain rule: $\frac{d}{dx} (\ln(\ln(\ln (x)))) = \frac{1}{(\ln(\ln(x)))} * \frac{1}{\ln(x)} * \frac{1}{x}$

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