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

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

Sep 26, 2016

$0 < x < e$

#### Explanation:

Calling $y = \frac{x}{\ln x}$, $y$ is decreasing for $\forall x | \frac{\mathrm{dy}}{\mathrm{dx}} < 0$

but $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{\ln} x \left(1 - \frac{1}{\ln} x\right)$. Now calling $z = \frac{1}{\ln} x$ we have the condition

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

$0 < \frac{1}{\ln} x < 1$ or $0 < x < e$