Prove without using a calculator $(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}<\pi$ Prove without using a calculator
$$(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}<\pi$$
I want to know if there is an easy way to prove this inequality without using a calculator.
 A: I don't know how to prove it, yet. However, I can help motivate a reason to solve it.
As to your question, 
"I want to know if there is an easy way to prove this inequality without using a calculator?"
The answer is no. There is no easy/quick proof of this that will make it's way into an answer. However, I have made some observations.
We have a recursion relation of the following,
$$L(x+1)=\ln(x+1)^{L(x)}$$
$$L^{(0)}(\ln(x))=\ln(x) \quad L^{(1)}(\ln(x))=\ln(x+1)^{\ln(x)}$$
$$\ln(23)=3.13549...$$
$$L^{(1)}(\ln(6))={\ln(7)}^{\ln(6)}=3.29623...$$
$$L^{(2)}(\ln(4))={\ln(6)}^{{\ln(5)}^{\ln(4)}}=\color{green}{3.08961...}$$
$$L^{(3)}(\ln(3))={\ln(6)}^{{\ln(5)}^{{\ln(4)}^{\ln(3)}}}=\color{blue}{3.16664...}$$
$$L^{(4)}(\ln(2))={\ln(6)}^{{\ln(5)}^{{\ln(4)}^{{\ln(3)}^{\ln(2)}}}}=\color{red}{3.14157...}$$
$$L^{(5)}(\ln(1))={\ln(6)}^{{\ln(5)}^{{\ln(4)}^{{\ln(3)}^{\ln(2)^{\ln(1)}}}}}=\color{blue}{3.16664...}$$
$$L^{(6)}(\ln(0))={\ln(6)}^{{\ln(5)}^{{\ln(4)}^{{\ln(3)}^{\ln(2)^{\ln(1)^{\ln(0)}}}}}}=\color{green}{3.08961...}$$
I find that interesting and motivating.
Now, some of you might be intrigued by colorful use of $\ln(0)$ above. I actually found that using a limit, we observe the behavior of $L^{(6)}(\ln(x))$ as $x$ approaches $0$.
