What is the domain of the function $y=\ln (x+2)$?

A. $x<-2$
B. $x>-2$
C. $x<2$
D. $x>2

The domain of the natural logarithm function ln ( u ) requires 0"> u > 0 .
For y = ln ( x + 2 ) , we need 0"> x + 2 > 0 .
Solving the inequality 0"> x + 2 > 0 gives -2"> x > − 2 .
The domain of the function is -2"> x > − 2 , so the answer is -2}"> x > − 2 ​ .

Explanation

Understanding the Problem We are asked to find the domain of the function y = ln ( x + 2 ) . The domain of a function is the set of all possible input values (x-values) for which the function is defined.

Logarithm Condition The natural logarithm function, denoted as ln ( u ) , is only defined for positive values of u . In other words, the argument of the natural logarithm must be greater than zero. Therefore, for the function y = ln ( x + 2 ) to be defined, we must have 0"> x + 2 > 0 .

Solving the Inequality To find the domain, we need to solve the inequality 0"> x + 2 > 0 for x . Subtracting 2 from both sides of the inequality, we get:


0 \implies x > -2"> x + 2 > 0 ⟹ x > − 2

Determining the Domain Therefore, the domain of the function y = ln ( x + 2 ) is all real numbers x such that -2"> x > − 2 .

Examples
The logarithm function is used to model many real-world phenomena, such as the growth of populations, the decay of radioactive materials, and the intensity of earthquakes. For example, the Richter scale, which measures the magnitude of earthquakes, is a logarithmic scale. Understanding the domain of logarithmic functions is important for interpreting these models and making accurate predictions.

Answered by GinnyAnswer | 2025-07-05