Problem 76 - College Algebra

Question

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

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\log _{5}(x+6)\) is \((-6, +\infty)\).

Step 1: Analyze the function

First, note that the function is logarithmic, with base 5 and the argument is \(x+6\). Logarithms are only defined for positive numbers. This means that the argument of the logarithm, \(x+6\), has to be greater than 0.

Step 2: Find the domain

To find the domain of the function, calculate the values of \(x\) for which the argument \(x+6\) is greater than 0. To do so, solve the inequality \(x+6 > 0\). Subtract 6 from both sides of the inequality to isolate \(x\), to get the result \(x > -6\).

Step 3: Write the domain

Now that the condition for \(x\) is found, you can describe the domain. The domain of the function \(f(x)=\log _{5}(x+6)\) is all \(x\) such that \(x > -6\). In interval notation, this is represented as \((-6, +\infty)\).

Key Concepts

Understanding Domains in Logarithmic Functions

The domain of a function represents all possible input values (or 'x' values) that allow the function to output real numbers. In the context of logarithmic functions, understanding the domain is crucial because logarithms are only defined for positive inputs. This means that for an expression like \(\log _{5}(x+6)\), the argument \(x+6\) must be greater than zero. If it's not, the logarithm will be undefined.

To find the domain of the function \(f(x)=\log _{5}(x+6)\), one must determine the set of x-values that keep the argument \(x+6\) positive. Solving the inequality \(x+6 > 0\) finds these values, leading us to conclude that \(x > -6\). This simple step ensures that the input to the logarithm is always valid, thus defining the domain of the function.

Solving Inequalities in Domains

Inequalities are mathematical statements used to express the relative size or order of two values. Solving inequalities is a critical skill for determining the domain of logarithmic functions.

For \(f(x)=\log _{5}(x+6)\), we solve the inequality \(x+6 > 0\). Simplifying inequalities involves performing the same operation on both sides. Subtracting 6 from both sides of the inequality gives \(x > -6\). Once simplified, this inequality tells us all the values of \(x\) that will keep the argument of the logarithm positive.

Inequalities not only help us find domains but are also essential in many areas of algebra. Understanding how to manipulate them opens up many pathways in mathematical problem-solving.

Expressing Domains Using Interval Notation

Interval notation is a concise way of writing subsets of the real numbers, showing the start and end point of the interval. When finding the domain of a function, interval notation provides a clear and standardized format to represent the solution.

From the inequality \(x > -6\), we know the domain of \(f(x)=\log _{5}(x+6)\) is all numbers greater than -6. Using interval notation, this is expressed as \((-6, +\infty)\).

This format uses parentheses \(()\) to indicate that -6 is not included in the solution set, known as being an "open interval". Plus, \(+\infty\) indicates that there is no upper bound, so the interval extends infinitely to the right on a number line. Interval notation thus is a powerful tool in conveying which values are included in the domain of a function in a direct and universal manner.