What values of x satisfies the following inequality :
${log}_{2} (2 x + 3) > {log}_{2} (3 x)$ ?

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Solution

Since the base is 2, which is greater than 1, the fact that ${log}_{2} (2 x + 3) > l o g_{2} (3 x)$ implies that $2 x + 3 > 3 x$ .
Subtracting $2 x$ from both sides gives $3 > x$ .
Additionally, the arguments of both the logarithms must be positive, so $3 x > 0$ and $2 x + 3 > 0$ . The first is more restrictive, as $x > 0$ .
So, the final solution set is $0 < x < 3$ .

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