Show that $\frac 1{\log_2x}+\frac 1{\log_3x}+\cdots+\frac 1{\log_{43}x}=\frac 1{\log_{43!}x}$.I am just not able to get it.please help.
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Can you express $\log_b x$ in terms of the natural logarithm? That would help a lot. – Daniel Fischer Jun 20 '14 at 09:48
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@DanielFischer Is the question not possible without natural logarithm? – Snehil Sinha Jun 20 '14 at 09:50
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Can you show that $(\log_ab)(\log_ba)=1$? – Gerry Myerson Jun 20 '14 at 09:53
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@GerryMyerson yes i can – Snehil Sinha Jun 20 '14 at 09:54
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I suppose you could do it with the logarithm of any base (take $2$ for example) – Claude Leibovici Jun 20 '14 at 09:55
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You could also use the logarithm to the base $123456$, $37$, or $10$ as a unifying thing. But the trick is to express everything in the same base. – Daniel Fischer Jun 20 '14 at 09:55
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Then do you see how to use that formula to answer the question? – Gerry Myerson Jun 20 '14 at 09:56
