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If the base of the logarithm is e, one can say log(x)/x takes maximum at e.

If the base of the logarithm is 10, one can say log(x)/x takes maximum at 10.

But log10(x)/x is nothing but (loge(e)/loge(10))/x.

The two functions are just a constant multiple (1/loge(10)) of each other. Shouldn't they have the same maxima?

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    $\begingroup$ You are right, unless we do something grotesque and take a base in the interval $(0,1)$. The maximum is at the same place. Of course it does not have the same value. $\endgroup$
    – André Nicolas
    Commented Oct 7, 2013 at 16:37
  • $\begingroup$ @AndréNicolas, but doesn't my last statement contradict everything? $\endgroup$
    – learner
    Commented Oct 7, 2013 at 16:45
  • $\begingroup$ Also, can someone please help me with tags here? I can't seem to add any relevant tags. None among "minima", "extrema", "decreasing" exist. $\endgroup$
    – learner
    Commented Oct 7, 2013 at 16:46
  • $\begingroup$ Oh, I didn't read your second sentence, which is wrong. They all are max at $x=e$. $\endgroup$
    – André Nicolas
    Commented Oct 7, 2013 at 16:47

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Your last sentence is correct, whatever base $b$ we take (as long as $b\gt 1)$, the function $\frac{\log_b x}{x}$ reaches a maximum at $x=e$. The reasoning that led to that conclusion is clearly stated.

The assertion in the second sentence that $\frac{\log_{10} x}{x}$ reaches a maximum at $x=10$ is not true.

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