What is the domain of $y=\log _4(x+3)$?
A. all real numbers less than -3
B. all real numbers greater than -3
C. all real numbers less than 3
D. all real numbers greater than 3
Answer (1)
The domain of a logarithmic function requires its argument to be positive.
Set up the inequality 0"> x + 3 > 0 .
Solve the inequality to find -3"> x > − 3 .
The domain is all real numbers greater than -3, which is all real numbers greater than -3 .
Explanation
Understanding the Logarithmic Domain We are asked to find the domain of the function y = lo g 4 ( x + 3 ) . The domain of a logarithmic function is the set of all real numbers for which the argument of the logarithm is positive. In this case, the argument is x + 3 .
Setting up the Inequality For the logarithm to be defined, we require that the argument is strictly greater than zero. Therefore, we need to solve the inequality: 0"> x + 3 > 0
Solving the Inequality Subtracting 3 from both sides of the inequality, we get: -3"> x > − 3
Determining the Domain This means that the domain of the function y = lo g 4 ( x + 3 ) is all real numbers greater than -3.
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and modeling population growth or decay. Understanding the domain of a logarithmic function is crucial in these contexts because it tells us the valid range of inputs for the model. For example, if we are modeling the population growth of a species using a logarithmic function, the domain tells us the minimum initial population size for the model to be meaningful.
