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I am trying to know which one is bigger :$$\log_9 71$$ or $$\log_8 61$$ how can i know without using a calculator ?

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    This may not be very scientific, but in this case I would just say $9^2 = 81$ is "relatively further off" from $71$ than $8^2 = 64$ is from $61$. So $\log_9 71 < \log_8 61 < 2$.2012-11-02
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    @ TMM - what's the break even point? Replacing $9$ by $a$, $71$ by $b$, $8$ by $c$ and $61$ by $d$, what's a nontrivial condition on $a$, $b$, $c$ and $d$ such that one will have inequality in one direction?2012-11-02
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    @Jonah: That's why I said it's not "very scientific" and why I used quotes for "relatively further off". It's more of a gut feeling ("the small difference between $a$ and $c$ does not compensate for the big difference between $b$ and $d$") than a careful analysis.2012-11-02
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    @ern: Was this a question in an algebra-precalculus course? Because this is quite difficult, even *with* the use of calculus.2012-11-02
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    @ TMM - I was just raising the question, wasn't criticizing your answer.2012-11-03

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