Find the minimum of an expression.

Question

$\begin{array}{l}Say\;x_1,x_2,x_3\in(0,1)\cup(1,+\infty)\\Find\;the\;minimum\;of\;the\;expression:\\E=\log_{x_1}\left(x_2x_3\right)+\log_{x_2}\left(x_1x_3\right)+\log_{x_3}\left(x_1x_2\right)\end{array}$

I literally don't have any idea.

Answer 1

Remember that $$\log_a b = {\log b\over \log a}$$

If we use arithmetic mean - geometric mean $$ {a_1+...+a_6 \over 6}\geq\sqrt[6]{a_1...a_6 } $$

Suppose that all $x_i<1$ or all $x_i>1$, then we get:

$$E=\log_{x_1}x_2 +\log_{x_1} x_3+\log_{x_2}x_1 +\log_{x_2}x_3+\log_{x_3}x_1+ \log_{x_1}x_2 \geq 6$$

---

If two of $x_i$ are $>1$ and one $<1$ (or vice versa) we get something like this:

If $x_1 = x_3=2$ and $0<x_3=x<1$ arbitrary we have:

$$ E = 2t+{2\over t}+2$$ where $t=\log _2x \in (-\infty ,0)$. So as $t\to -\infty$ we have $E\to -\infty$.