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Which is bigger among (i) $\log_2 3$ and $\log _3 5$ (ii) $\log_2 3$ and $\log _3 11$.

Trial: I use AM-GM inequality and get $$\dfrac{\log_e 2+\log_e 5}{2}>\sqrt{\log_e 2\cdot \log_e 5} \\ \implies \dfrac{\log _e10}{2}>\sqrt{\log_e 2\cdot \log_e 5} $$ Then I can not proceed. Please help.

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  • $\begingroup$ (i) & (ii) are two independent questions right? $\endgroup$ – lab bhattacharjee Jun 9 '13 at 14:08
  • $\begingroup$ @labbhattacharjee : Yes. You are right. $\endgroup$ – A.D Jun 9 '13 at 14:09
  • $\begingroup$ What about the change of base property $log_ab = \frac{1}{log_ba}$? $\endgroup$ – Wortel Jun 9 '13 at 14:27
  • $\begingroup$ @Wortel How can I use this property to solve the problem? $\endgroup$ – A.D Jun 9 '13 at 14:29
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(i) Since $5^2=25\lt27=3^3$ and $3^2=9\gt8=2^3$, $2\log_35\lt3\lt2\log_23$ hence $\log_35\lt\log_23$.

(ii) Since $3\lt4=2^2$ and $11\gt9=3^2$, $\log_23\lt2\lt\log_311$.

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For the (ii) $\log_311>\log_39=\log_3(3^2)=2$ but $\log_23<\log_24=\log_2(2^2)=2$

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