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Basics & Logarithms: Number Natural Numbers Number 14x Defined

JEE Maths question with a full step-by-step solution.

Question
Number of natural numbers for which the number log⁡4−x(x2−14x+45)\log_{4-x}(x^2-14x+45) is defined is:
A44
B33
C22correct
D11
Solution
Step 1: Conditions for the logarithm to be defined Argument: x2−14x+45>0  ⟹  (x−9)(x−5)>0  ⟹  x<5x^2-14x+45 > 0 \implies (x-9)(x-5) > 0 \implies x < 5 or x>9x > 9. Base: 4−x>04-x > 0 and 4−x≠1  ⟹  x<44-x \neq 1 \implies x < 4 and x≠3x \neq 3. Step 2: Apply to natural numbers Combining x<4x < 4 (which forces x<5x < 5) with x≠3x \neq 3 for natural xx:
x∈{1,2}x \in \{1, 2\}
 There are two natural numbers. Answer: (3)
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