Problem 75 - Algebra and Trigonometry

Question

In Exercises 75–80, find the domain of each logarithmic function. $$ f(x)=\log _{5}(x+4) $$

Step-by-Step Solution

Answer

The domain of the given logarithmic function is \((-4, \infty)\).

Step 1: Define the inequality

For the logarithmic function \(f(x)=\log_{5}(x+4)\) to be defined, the expression \(x+4\) must be greater than zero. We write this as an inequality: \(x+4 > 0\).

Step 2: Solve the inequality

We solve the inequality to determine the range of \(x\) where the function is defined. This is done by subtracting 4 from both sides of the inequality: \(x > -4\).

Step 3: State the domain of the function

The domain of the function \(f(x)=\log_{5}(x+4)\) is thus all real numbers greater than -4. This can be written in interval notation as \((-4, \infty)\).

Key Concepts

Logarithmic Functions

Understanding logarithmic functions is crucial when determining the domain of any log-based expression. The essence of a logarithmic function lies in its ability to "undo" or reverse an exponential function. For a given logarithmic function, say \( f(x) = \log_b(a) \), \( b \) is the base of the logarithm, and \( a \) is the argument. The fundamental rule for any logarithmic function to exist is that the argument \( a \) must always be greater than zero. This is because logarithms are only defined for positive positive numbers. Some key characteristics of logarithmic functions are:

• Their domain (the set of all input values \( x \)) is restricted to positive values of their arguments.
• They have a vertical asymptote at \( x = 0 \) since the function increases rapidly without actually reaching zero.
• Their range is all real numbers, denoted by \((-\infty, \infty)\).

With these ideas in mind, when working with a logarithmic function like \( f(x)=\log_{5}(x+4) \), always begin by ensuring the expression inside the logarithm, \( x+4 \), is positive by setting it greater than zero, leading us to solve an inequality.

Inequalities

To work with inequalities is to determine the conditions under which a certain mathematical statement holds true. In the context of logarithmic functions, solving an inequality helps in defining the domain. For the function \( f(x)=\log_{5}(x+4) \), we need \( x+4>0 \). This simply means :

• We take the expression inside the logarithm, \( x+4 \), and set it greater than zero.
• This leads us to reorder the terms logically for clarity, resulting in \( x > -4 \).

Consider inequalities as a way to narrow down the possible values of \( x \) that satisfy the logarithm's condition. It is important as it paves the way to express the domain of the function in a way that is universally understood.Through solving the inequality, we found the domain consists of numbers larger than \(-4\). Proper manipulation and understanding of inequalities are thus pivotal for defining the scope of logarithmic functions.

Interval Notation

Interval notation is a systematic method of denoting the set of numbers between two endpoints, often capturing the domain of functions concisely. In interval notation:

• Parentheses \((\) and \()\) are used to denote numbers that are not included in the interval.
• Square brackets \([\) and \()]\) are used to indicate numbers that are included.

For the logarithmic function \( f(x)=\log_{5}(x+4) \), having solved the inequality \( x > -4 \), the domain of the function in interval notation is expressed as \((-4, \infty)\). This notation signifies:

• \(-4\) is not included in the set, indicated by the parenthesis.
• The domain stretches infinitely in the positive direction, represented by \(\infty\).

This systematic way of expressing sets offers clear communication of the continuous stretch of all real numbers greater than \(-4\) as the values \( x \) can take.