Question

The value of $x$, which satisfies the inequation, $({\log }_{\left(1/2\right)}x\ge {\log }_{\left(1/3\right)}x)$ is

• A. $(0,1]$
• B. $(0,1)$
• C. $[0,1)$
• D. none

Answer 1

The correct option is A $(0,1]$

${\log }_{1/2}x\ge {\log }_{1/3}x$

We can write this as

$\Rightarrow \frac{\log x}{\log 1/2}\ge \frac{\log x}{\log 1/3}$

$\Rightarrow \frac{-\log x}{\log 2}\ge \frac{-\log x}{\log 3}$

$\Rightarrow \frac{\log x}{\log 2}\le \frac{\log x}{\log 3}$

$\Rightarrow \frac{\log x}{\log 2}-\frac{\log x}{\log 3}\le 0$

$\Rightarrow \log x(\frac{1}{\log 2}-\frac{1}{\log 3})\le 0$

$\Rightarrow \log x\left(\frac{\log 3-\log 2}{\log 2.\log 3}\right)\le 0$

$\Rightarrow \log x\left(\log \frac{3}{2}\right)\le 0$

As $\log \frac{3}{2}>0$ then $\log x\le 0$

This is only possible when $x$ belongs to $(0,1]$