# Where does log(x) / x take maximum value?

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?

• 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. – André Nicolas Oct 7 '13 at 16:37
• @AndréNicolas, but doesn't my last statement contradict everything? – learner Oct 7 '13 at 16:45
• Also, can someone please help me with tags here? I can't seem to add any relevant tags. None among "minima", "extrema", "decreasing" exist. – learner Oct 7 '13 at 16:46
• Oh, I didn't read your second sentence, which is wrong. They all are max at $x=e$. – André Nicolas Oct 7 '13 at 16:47

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.