# What are possible values of x if  lne^(2x)/2<xln(3)??

Nov 3, 2015

Question asked for 'values' so you have to state all of them.
x in ]0,oo] Notation for the range of zero to infinity but excluding 0

#### Explanation:

$\ln \left({e}^{2 x}\right)$ has the same value as $2 x \ln \left(e\right)$
$\ln \left(e\right) = 1$ so we now have:

$\frac{2 x}{2} < x \ln \left(3\right)$

$x < x \ln \left(3\right)$

$0 < x \ln \left(3\right) - x$

$0 < x \left(\ln \left(3\right) - 1\right)$

$\frac{0}{\ln \left(3\right) - 1} < x$

$x > 0$

Nov 3, 2015

This inequality is true for any positive real number x in (0;+oo)

#### Explanation:

$\ln \frac{{e}^{2 x}}{2} < x \ln \left(3\right)$

$\frac{2 x \ln e}{2} < x \ln 3$

$x < x \ln 3$

$x \left(1 - \ln 3\right) < 0$

$x > 0$

I changed the sign of inequality in last expression because $\ln 3 \approx 1.098$, so last step was the division by a negative number ($1 - \ln 3$)