Problem 55 - Precalculus

Question

State the domain of the logarithmic function in interval notation. $$f(x)=\log _{2}(x+5)$$

Step-by-Step Solution

Answer

The domain of \( f(x) = \log_2(x + 5) \) is \((-5, \infty)\).

Step 1: Understand the Function

The function provided is a logarithmic function, specifically of the form \( f(x) = \log_2(x + 5) \). The function \( \log_b(a) \) is only defined for \( a > 0 \). Therefore, we need to identify the values of \( x \) for which \( x + 5 > 0 \).

Step 2: Set the Inequality

For the logarithmic function to be defined, solve the inequality \( x + 5 > 0 \). This inequality represents the condition under which the expression inside the logarithm, \( x + 5 \), remains positive.

Step 3: Solve the Inequality

Solve \( x + 5 > 0 \) by isolating \( x \). Subtract 5 from both sides of the inequality to get \( x > -5 \).

Step 4: Express the Domain in Interval Notation

The domain consists of all the values of \( x \) that satisfy \( x > -5 \). In interval notation, this is expressed as \( (-5, \infty) \).

Key Concepts

Domain

The domain of a function is the set of all possible input values (usually represented by the variable \( x \)) that the function can accept without running into any mathematical issues. For a logarithmic function like \( f(x) = \log_2(x + 5) \), the primary concern is ensuring that the argument of the logarithm is positive. This is because the logarithmic function is undefined for zero and negative numbers.
For the function \( \log_2(x + 5) \), this means that \( x + 5 \) must be greater than zero. Solving the inequality \( x + 5 > 0 \) helps us find the domain, which represents all \( x \)-values fulfilling this condition. Understanding the domain is crucial as it sets the boundaries for successful and meaningful computation of the function values.

Interval Notation

Once you have determined the range of acceptable \( x \)-values, interval notation is used to clearly express the domain. Interval notation uses brackets and parentheses to indicate the set of values that are part of the domain.
For instance, if we find that \( x > -5 \) is the requirement for \( f(x) = \log_2(x + 5) \), then the domain in interval notation would be \( (-5, \infty) \).

• The round parenthesis "(" indicates that \(-5\) is not included in the set (this reflects the inequality \( x > -5 \)).
• The infinity symbol "\( \infty \)" and its parenthesis mean that the domain extends indefinitely in the positive direction.

Interval notation provides a clean and precise way to describe the domain, helping avoid any confusion about which values are part of the domain.

Inequality Solving

Solving inequalities is a fundamental algebraic technique essential for finding the domain of many functions, including logarithmic ones. In this process, you manipulate the inequality to isolate the variable on one side.
In the given function \( f(x) = \log_2(x + 5) \), we start with the inequality \( x + 5 > 0 \). To solve this, follow these steps:

• Subtract 5 from both sides of the inequality, yielding \( x > -5 \).

The solution \( x > -5 \) is simple but crucial, as it determines the entire domain for which the function is defined. Understanding how to solve such inequalities helps in analyzing and designing many real-world systems where logarithmic functions are used. Always check your inequality solutions by plugging values back into the original equation to ensure they satisfy the function's conditions.