What is the domain of the function $y=\ln (x+2)$ ?
The function given is \(y = \ln(x + 2)\). To find the domain of this function, we must consider the definition of the natural logarithm function. The expression \(\ln(u)\) is defined only when its argument \(u\) is strictly greater than zero, i.e., \(u > 0\). In our function, the argument is \(x + 2\).
Since the natural logarithm requires its argument to be strictly positive, we set up the following inequality:
\[x + 2 > 0\]
To solve for \(x\), we subtract 2 from both sides of the inequality:
\[x > -2\]
Thus, the function is defined for all values of \(x\) that are greater than \(-2\).
The domain of the function \(y = \ln(x + 2)\) is all \(x\) such that \(x > -2\). Comparing this result with the provided options:
The correct option is (c). Therefore, the domain of the function is \(\boxed{x > -2}\).
In real-world applications, logarithmic functions are used to model phenomena that grow or decay over time, such as population growth, the spread of a virus, or the intensity of sound (decibels). For instance, if you are modeling the growth of a colony of bacteria using the function \(f(t) = \ln(t + 2)\) where \(t\) represents time in hours, the domain \(t > -2\) ensures that the mathematical model is only applied to physically meaningful time intervals. Even though the mathematical domain includes negative values greater than -2, the physical application would further constrain the domain to \(t \geq 0\). Understanding domain restrictions is crucial for ensuring that mathematical models correctly represent physical reality.