The set of positive real values of x for which the function $f(x) = x/(ln⁡x)$ is a decreasing function is ?

Question

a) $x < e$
b) $x = 1$
c) $x < e^2$
d) $x > e$
e) empty space

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Answer 1 · 2016-09-26T16:09:29

$0 < x < e$

Calling $y = x/(ln x)$ , $y$ is decreasing for $forall x | dy/(dx) < 0$

but $dy/(dx) = 1/ln x(1-1/lnx)$ . Now calling $z = 1/lnx$ we have the condition

$z(1-z) < 0$ and this happens for $0 < z < 1$ so $y$ is decreasing for

$0 < 1/lnx < 1$ or $0 < x < e$