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Find the minimum value of $$\frac{\log_bc}{\log a} + \frac{\log_ca}{\log b} + \frac{\log_ab}{\log c}$$

since I do not know how to write log base a index a so I gave it in that manner. I tried out the problem and thought intuitively that the answer should be $1$ but cant find properly any reason.

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  • $\begingroup$ upper log is like this $\log_b c$ and what is the base in denominator $\endgroup$ – iostream007 Jul 28 '13 at 11:50
  • $\begingroup$ Is anything else given? Like $\log a, \log b, \log c > 0$ or something? $\endgroup$ – Parth Thakkar Jul 28 '13 at 11:58
  • $\begingroup$ @iostream007, why did you edit it this way? I am not sure what you wrote is correct. As for the the base is, it must be $e$ or $10$. But does it matter? $\endgroup$ – Parth Thakkar Jul 28 '13 at 11:59
  • $\begingroup$ @soumyajit Am I edit it correctly? $\endgroup$ – iostream007 Jul 28 '13 at 12:00
  • $\begingroup$ @Soumajit, if you don't reply to any questions asked, obviously no one is gonna bother to answer your question. $\endgroup$ – Parth Thakkar Jul 28 '13 at 12:04
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The expression is the same as $$\frac{\log(c)}{\log(a)\log(b)}+\frac{\log(a)}{\log(b)\log(c)}+\frac{\log(b)}{\log(a)\log(c)}$$

Fix $a$ and $b$, and make $c$ tend to $0$. What do you get?

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