Math Inequality using induction?

Question

Prove that $\log_3\pi + \log_\pi 3 > 2$ without using log tables.

I was thinking of using strong induction for something like this, but I find it a difficult thing to come by, especially giving the lack of variables. How does one generally approach a problem like this?

Thanks!

Answer 1

Clearly $\pi\ne3$, so $$(\ln\pi-\ln3)^2>0\ .$$ Expanding and taking the middle term to the RHS, $$(\ln\pi)^2+(\ln3)^2>2(\ln\pi)(\ln3)\ .$$ Dividing both sides by $(\ln\pi)(\ln3)$ gives $$\frac{\ln\pi}{\ln3}+\frac{\ln3}{\ln\pi}>2\ ,$$ and the "change of base" formula gives the desired result.

Answer 2

By using change of base formula and AM-GM inequality we have $$\log_3\pi+\log_{\pi}3=\frac{\ln \pi}{\ln 3}+\frac{\ln 3}{\ln \pi}> 2\sqrt{\frac{\ln \pi}{\ln 3}\frac{\ln 3}{\ln \pi}}=2$$