What is the domain of the function $y=\ln (x+2)$?
To determine the domain of the function \(y = \ln(x + 2)\), we must identify the set of all real numbers \(x\) for which the function is defined. The natural logarithm function, \(\ln(u)\), is defined if and only if its argument \(u\) is strictly greater than zero, i.e., \(u > 0\). In the given expression \(y = \ln(x + 2)\), the argument is \(u = x + 2\).
Based on the definition of the natural logarithm, we require the argument \(x + 2\) to be strictly greater than zero:
\[x + 2 > 0\]
To solve for \(x\), we subtract \(2\) from both sides of the inequality:
\[x > -2\]
This inequality defines all possible values of \(x\) for which the function \(\ln(x + 2)\) exists. Therefore, the domain of the function is all real numbers \(x\) such that \(x > -2\).
We have found the domain to be \(x > -2\). Comparing this result with the given options:
Our result matches option a. Thus, the domain of the function \(y = \ln(x + 2)\) is \(x > -2\). The final answer is \(\boxed{a}\).
Understanding the domain of logarithmic functions is essential in various real-life scenarios, such as modeling population growth, decay, or financial interest rates. For instance, if you are modeling the growth of a bacterial colony over time \(t\) using a function like \(N(t) = C \cdot \ln(t + 2)\), you need to know the valid range for \(t\) (time) to make meaningful predictions. In this case, since the argument \(t + 2\) must be positive, it ensures that your model is only applied to values of time that are logically consistent with the natural logarithm's definition, avoiding invalid calculations in your simulation or analysis.