Let this number be $B$, $B$ achieves the following:
$\frac{\ln(x)}{\ln(B)}>0$, for x>1
$\frac{\ln(x)}{\ln(B)}<0$, for 1>x>0
If such a number is worth serious attention, what is its name by which I can find more information?
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I'm quite confused after reading your question - not sure exactly what you're saying. Could you rewrite it more clearly? – Aurey Jun 03 '16 at 01:51
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Is it the formatting that is the problem? – user344265 Jun 03 '16 at 01:52
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All logarithmic functions with a base greater than one satisfy the description.
$$\log_B{x}=\frac{\log{x}}{\log{B}}$$
For all real values of $B$ and $B>1$, the value of $\log{B}$ is greater than zero and finite.
The function $\log{x}$ is greater than zero for all $x>1$, and negative for all $0<x<1$
This set of output does not change in positivity/negativity by the division of a real, positive denominator, therefore, all real values of $B$ greater than one satisfy the descriptions.
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