The smallest prime number $x$ , satisfying $l o g_{2 x} (x^{2} - 5 x + 6) > 1$ , is

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Solution

We have,
${log}_{2 x} (x^{2} - 5 x + 6) > 1$
$f (x) = l o g_{2 x} (x^{2} - 5 x + 6)$

At $x = 2, 3, 5, w e g e t f (x) < 1$
When $x = 7,$
$f (7) = l o g_{2 \times 7} (7^{2} - 5 \times 7 + 6)$
$f (7) = l o g_{14} 20$
$f (7) = 1.53$
$f (7) > 1$

Hence, the smallest prime number is $7$ .

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