If $ \log \log x =1$, then is it true that $e^e =x$?

Question

I worked out this question, and I wanted to see if my understanding of the concepts involved is sound.

Solve for $x$ $$\ln(\ln(x))=1$$

$$e^1=\ln(x)$$ $$e^e=x$$ Since any number raised to $1$ is just itself, the final answer could be expressed as $x=e^e$ This is not a homework question, but a question I stumbled across which I find interesting: I want to see if my understanding is correct.

You could've checked that $\log e^e = e\log e=e$, and then $\log e =1$, which is what you want. — Pedro, Apr 29 '12 at 23:47

score 3 · Accepted Answer · answered Apr 29 '12 at 23:49

3

It is correct. To verify, plug the answer back into the equation and check that both sides are equal:

\begin{align*} \ln(\ln(e^e)) = \ln(e \ln(e)) = \ln(e \cdot 1) = \ln(e) = 1 \end{align*}

Remember that $\ln(a^b)=b\ln(a)$ and $\ln(e) = 1$.

answered Apr 29 '12 at 23:49

Ayman Hourieh

39,603

is that $\ln(a^b)=b\ln(a)$ correct !!? for $b\in \mathbb{R}-\mathbb{Q}$ – Abdelmajid Khadari Apr 30 '12 at 00:46
@AbdelmajidKhadari Yes. It is a direct consequence of the definition of the logarithm. – Pedro Jul 06 '12 at 06:12

If $ \log \log x =1$, then is it true that $e^e =x$?

1 Answers1