Problem 48 - Introductory and Intermediate Algebra for College Students 4th

Question

Find the domain of each logarithmic function. $$f(x)=\log _{5}(x+6)$$

Step-by-Step Solution

Answer

The domain of the logarithmic function is \(-6 < x < ∞\), or in interval notation, the domain is \((-6, ∞)\)

Step 1: Isolate the Argument of the Logarithm

We start with the inequality \(x+6>0\), as the parentheses in the logarithm expression gives us the range we need. The argument of the logarithm, that is the expression inside the parentheses, needs to be greater than 0.

Step 2: Solve the inequality

Subtract 6 from both sides of the inequality to isolate x: \(x + 6 - 6 > 0 - 6\), simplifying it gives: \(x > -6\)

Step 3: Write the domain

The domain of the function is all x such that \(x > -6\). In interval notation, this can be written as (-6, +∞). This is the set of all real numbers greater than -6.

Key Concepts

Domain of a Function

When dealing with logarithmic functions, it's essential to determine where the function is defined. This area is known as the domain of a function. For a logarithmic function like \(f(x) = \log_5(x+6)\), the domain represents all the possible values of \(x\) for which the function is valid and returns a real number.

Logarithmic functions have a crucial requirement: the argument (the expression inside the log) must be greater than zero. Since a logarithm is not defined for zero or negative numbers, we need to ensure the argument satisfies:

• \(x+6 > 0\)
• Subtracting 6 from both sides, we get \(x > -6\)

This means that the domain for \(f(x) = \log_5(x+6)\) is all real numbers greater than \(-6\).

By understanding and finding the domain, you ensure that the function behaves correctly and you avoid undefined scenarios.

Inequalities

Inequalities are mathematical expressions used to represent the relationship of one quantity being less than, greater than, or not equal to another. Solving inequalities is similar to solving equations, but you have to be mindful of the direction of the inequality sign when adding, subtracting, multiplying, or dividing both sides.

For the function \(f(x) = \log_5(x+6)\), the inequality \(x+6 > 0\) ensures the argument of the logarithm is positive. Here’s how you solve it:

• Isolate the variable: Begin by moving any constants on the same side as \(x\). For \(x + 6 > 0\), subtract 6 from both sides to get \(x > -6\).
• Keep the inequality direction: When you add or subtract, the inequality sign will stay the same. Multiplication or division by a negative number, however, would flip the sign.

Understanding inequalities helps in defining the domain of functions and in solving real-world problems where conditions need to be met.

Interval Notation

Interval notation is a way to describe a set of numbers between two endpoints without listing all possible values. This simplification is particularly useful when writing the domain of a function.

To represent the domain of \(f(x) = \log_5(x+6)\) which is \(x > -6\), we use interval notation: (-6, +∞). Here's how it works:

• Round brackets \(()\) indicate that the endpoint is not included, known as an open interval.
• The interval \((-6, +∞)\) means all numbers greater than \(-6\) but not including \(-6\) itself, extending to infinity.
• The symbol \(∞\) (infinity) always uses a round bracket as it represents an unbounded limit that cannot be "reached".

Interval notation provides a compact, understandable way to express domains and ranges in mathematics.