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Conditions on the Parameters of Logarithm Function

What values o...

Question

What values of x satisfy the following inequality ?
${log}_{1 / 2} (3 x) > {log}_{1 / 2} (2 x + 3)$ ?

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Solution

Since the base is $\frac{1}{2}$ , which is less than 1, the given inequality implies $3 x < 2 x + 3$ . Then, $x < 3$ .
Additionally, the argument of each logarithm must be positive, so $3 x > 0$ and $2 x + 3 > 0$ .
The former is more restrictive, so the solution set is $0 < x < 3$ .

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