First slide

Introduction to real valued functions

Question

$Solution set of inequality \log_{1 / 2} x + \log_{3} x > 1 is$

Moderate

A

$x < 2^{\frac{1}{\log_{3} 2 - 1}}$
B

$x < 3^{\frac{1}{\log_{3} 2 - 1}}$
C

$x > 2^{\frac{1}{\log_{3} 2 - 1}}$
D

$x < 3^{\frac{1}{\log_{2} 3 - 1}}$

Solution

$\begin{array}{l} \log_{1 / 2} x + \log_{3} x > 1 \\ \Rightarrow - \log_{2} x + \log_{3} x > 1 \\ \Rightarrow \log_{3} x > 1 + \log_{2} x \\ \Rightarrow \frac{\log_{2} x}{\log_{2} 3} > 1 + \log_{2} x \\ \Rightarrow (\frac{1}{\log_{2} 3} - 1) \log_{2} x > 1 \\ \Rightarrow \log_{2} x < \frac{1}{\log_{3} 2 - 1} \\ \Rightarrow 0 < x < 2^{\frac{1}{\log_{3} 2 - 1}} \end{array}$

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