Prove without using a calculator $(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}<\pi$

Question

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.

Answer 1

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 $0$.