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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}(x^2-14x+45)

is defined is:

4

3

2

correct

1

Solution

Step 1: Conditions for the logarithm to be defined Argument:

x^2-14x+45 > 0 \implies (x-9)(x-5) > 0 \implies x < 5

x > 9

. Base:

4-x > 0

and

4-x \neq 1 \implies x < 4

and

x \neq 3

. Step 2: Apply to natural numbers Combining

x < 4

(which forces

x < 5

) with

x \neq 3

for natural

x

x \in \{1, 2\}

There are two natural numbers. Answer: (3)

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