A problem of Logarithm

Question

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.

upper log is like this $\log_b c$ and what is the base in denominator — iostream007, Jul 28 '13 at 11:50
Is anything else given? Like $\log a, \log b, \log c > 0$ or something? — Parth Thakkar, Jul 28 '13 at 11:58
@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? — Parth Thakkar, Jul 28 '13 at 11:59
@Soumajit, if you don't reply to any questions asked, obviously no one is gonna bother to answer your question. — Parth Thakkar, Jul 28 '13 at 12:04
@ParthThakkar In question he mentioned that he couldn't write base and index in log so he wrote in that way so I thought he wanted to write this. — iostream007, Jul 28 '13 at 12:08
Ok, then maybe I didn't understand him properly. But this is irritating, both of us are working at his problem and he isn't even interested. — Parth Thakkar, Jul 28 '13 at 12:11
@Soumajit, what are you intending to ask? please clarify, otherwise we can't answer. — Samrat Mukhopadhyay, Jul 28 '13 at 12:13
I think, there must be some constraints on $a,b,c$ to get some interesting answer. — Samrat Mukhopadhyay, Jul 28 '13 at 12:17
@iostream007 I suspect the intention was $\log_a (bc) + \log_b (ca) + \log_c (ab)$. That would lead to a classical minimisation problem. — Daniel Fischer, Jul 28 '13 at 12:29
@DanielFischer asker is not interested in replying so I'm not giving any effort — iostream007, Jul 28 '13 at 12:30
The OP may not have good Internet connection. Maybe has three kids and five cats and right now cannot reply. The good thing about this site is that your effort is not required for it to work. — OR., Jul 28 '13 at 12:42
the thing is log in which base is a and index contains bc simmillarly the others....\ — Soumajit Das, Jul 29 '13 at 17:14

score 1 · Answer 1 · answered Jul 28 '13 at 12:13

1

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?

answered Jul 28 '13 at 12:13

OR.

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A problem of Logarithm

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