Verified Answer :

  • Identify the requirement for the argument of the natural logarithm, which must be strictly greater than zero: \(x + 2 > 0\).
  • Solve the inequality to isolate \(x\): \(x > -2\).
  • Match the result with the given choices, confirming that option d is correct.
  • The final domain is \(\boxed{x > -2}\).

Explanation

  1. Problem Analysis and Domain Condition

To determine the domain of the function \(y = \ln(x + 2)\), we must consider the definition of the natural logarithm function. The natural logarithm \(\ln(u)\) is defined if and only if its argument \(u\) is strictly greater than zero, meaning \(u > 0\). In this specific function, the argument is \(u = x + 2\). Therefore, the function is defined only when the condition \(x + 2 > 0\) is satisfied.

  1. Solving the Inequality and Identifying the Correct Option

To find the domain, we solve the inequality for \(x\):
\[x + 2 > 0\]
Subtracting 2 from both sides of the inequality, we get:
\[x > -2\]
This means the function is defined for all real numbers \(x\) such that \(x\) is greater than \(-2\). Comparing this result with the provided options:

A\(x > 2\)
B\(x < 2\)
C\(x < -2\)
D\(x > -2\)

The result corresponds to option d.

Examples

In real life, logarithmic functions like \(y = \ln(x+2)\) are often used to model growth processes or signal processing. For instance, if an engineer is modeling the intensity of a sound or signal relative to a threshold, the argument of the logarithm must remain positive to represent a physical quantity. Knowing the domain of the function ensures that the model is only applied to valid inputs, preventing errors in calculations where the logarithm would be undefined (i.e., non-positive).

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