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Find the domain of definition of:

\(P\left( x \right) = \frac{x}{{\log \left( {2 + x} \right)}}\)
1. {x ∶ x ∈ R and x > 0 and x ≠ -2}
2. {x ∶ x ∈ R and x > -2 and x ≠ -1}
3. {x ∶ x ∈ R and x > -2 and x ≠ 0}
4. {x ∶ x ∈ R and x > -1 and x ≠ 2}

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Correct Answer - Option 2 : {x ∶ x ∈ R and x > -2 and x ≠ -1}

\(P\left( x \right) = \frac{x}{{\log \left( {2 + x} \right)}}\)

P(x) is defined, if log (2 + x) is real and also log (2 + x) ≠ 0.

Now,

1) log (2 + x) will be real if

2 + x > 0

x > -2

2) log (2 + x) ≠ 0 is satisfied if

2 + x ≠ 1

x ≠ -1

∴ The required domain of definition of P(x) is: 

{x ∶ x ϵ R and x > -2 and x ≠ -1}

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