What is the domain for the exponential function $f(x)=-4 \log (x+2)$?
D:$\left\{x \in R \right\}$
D:$\left\{x \in R \mid x>-2\right\}$
D:$\left\{x \in R \mid x>2\right\}$
D:$\left\{x \in R \mid x>4\right\}$
Answer (2)
The domain of a logarithmic function is the set of all x values for which the argument of the logarithm is positive.
Set the argument of the logarithm, x + 2 , greater than zero: 0"> x + 2 > 0 .
Solve the inequality for x : -2"> x > − 2 .
Express the domain in set notation: -2\}}"> D = { x ∈ R ∣ x > − 2 } .
Explanation
Understanding the Problem We are asked to find the domain of the function f ( x ) = − 4 lo g ( x + 2 ) . The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, we have a logarithmic function, and logarithms are only defined for positive arguments. Therefore, we need to find the values of x for which 0"> x + 2 > 0 .
Solving the Inequality To find the domain, we need to solve the inequality 0"> x + 2 > 0 . Subtracting 2 from both sides of the inequality, we get: -2"> x > − 2
Expressing the Solution This means that the function is defined for all real numbers x that are greater than − 2 . In set notation, this is written as -2\}"> D = { x ∈ R ∣ x > − 2 } .
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, modeling population growth, and calculating the pH of a solution. Understanding the domain of a logarithmic function is crucial in these applications because it tells us the range of valid input values for the model. For example, in the Richter scale, the magnitude of an earthquake is given by M = lo g 10 ( I / S ) , where I is the intensity of the earthquake and S is the intensity of a standard earthquake. Since the intensity I must be positive, the argument of the logarithm is always positive, ensuring that the magnitude M is defined.
The domain of the function f ( x ) = − 4 lo g ( x + 2 ) is -2 \}"> D = { x ∈ R ∣ x > − 2 } , meaning the function is defined for all real numbers greater than -2.
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