Problem 105 - Precalculus

Question

Explain the mistake that is made. State the domain of the logarithmic function \(f(x)=\log _{2}(x+5)\) in interval notation. Solution: The domain of all logarithmic functions is \(x>0\). Interval notation: \((0, \infty)\) This is incorrect. What went wrong?

Step-by-Step Solution

Answer

Domain is \((-5, \infty)\), not \((0, \infty)\) as calculated.

Step 1: Understand the Logarithmic Function

The logarithmic function given is \( f(x) = \log_{2}(x+5) \). A logarithmic function is defined only when the argument inside the logarithm is positive, i.e., \( x+5 > 0 \). Thus, we need to identify under what conditions \( x+5 \) is greater than zero.

Step 2: Find the Condition for the Positive Argument

To find when \( x+5 > 0 \), solve the inequality. Subtract 5 from both sides: \( x > -5 \). This inequality tells us that the function is defined for all \( x \) values greater than -5.

Step 3: Write the Correct Domain

In interval notation, the domain where \( f(x) = \log_{2}(x+5) \) is defined is \( (-5, \infty) \). This means \( x \) can take any value greater than -5.

Step 4: Identify the Mistake

The mistake in the solution provided was assuming the domain of the logarithmic function as \( x > 0 \). This assumption fails to consider the +5 inside the logarithm argument, which shifts the entire requirement for \( x \) five units to the left.

Key Concepts

Understanding Inequalities

When dealing with inequalities, it is crucial to understand how they describe a range of possible values for a variable. In the context of logarithmic functions, inequalities help us determine the set of inputs for which the function is defined.

For the logarithmic function \( f(x) = \log_{2}(x+5) \), we need the argument \( x+5 \) to be greater than 0. To set up the inequality, we write:

• \( x+5 > 0 \)

Solving this inequality by subtracting 5 from both sides gives us \( x > -5 \). This tells us that the domain for this function is all values of \( x \) that are greater than -5.

Inequalities often involve simple arithmetic operations such as addition or subtraction to isolate the variable. Once the inequality is solved, we can infer the range of valid inputs for a given function. Understanding how to set up and solve inequalities is essential in finding the domain of functions, especially those involving logarithms.

Interval Notation Explained

Interval notation is a shorthand used to describe a set of numbers along the number line. It is often used to express the domain or range of a function in a concise and clear manner.

In interval notation, \( (-5, \infty) \) represents all numbers greater than \(-5\). Here's how we break it down:

• The parenthesis \((\) next to \(-5\) means that \( -5 \) is not included in the set.
• The comma separates the two endpoints of the interval.
• \( \infty \), meaning infinity, suggests the interval extends without bound, indicating all numbers greater than \(-5\) are included.

This specific notation gives us a quick visual cue that \( f(x) = \log_{2}(x+5) \) excludes \(-5\) itself, hence why the parentheses are used instead of a bracket. The simplicity and clarity of interval notation make it a valuable tool when conveying domains and ranges.

Insights into Logarithmic Functions

Logarithmic functions, such as \( f(x) = \log_{2}(x+5) \), are the inverses of exponential functions. Understanding their behavior is crucial in determining when and where they are defined.

One important feature of logarithmic functions is that their argument must be positive. This is because a logarithm represents the power to which a base number must be raised to produce a given number, and negative or zero values do not correspond to valid outputs in base 2 logarithms.

To ensure the argument is positive:

• Identify the terms inside the logarithm, such as \( x+5 \).
• Set up an inequality, \( x+5 > 0 \), which arises from the requirement for the logarithm's argument to be positive.
• Solve the inequality to find the conditions on \( x \).

By solving \( x > -5 \), we find that the theorem holds for all \( x > -5 \), which defines the domain of our logarithmic function. Understanding this process allows you to tackle similar exercises with ease, as it underscores the fundamental requirements for the functioning of logarithms.