Problem 76 - Algebra and Trigonometry

Question

In Exercises 75–80, 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 \(x > -6\).

Step 1: Identify the Argument of the Logarithm

The argument of the logarithm here is \(x+6\) inside \(\log_{5}(x+6)\).

Step 2: Set the Argument Greater Than Zero

To find the domain, it is needed to set up an inequality where the argument of the log function is greater than 0. So, in this case, \(x + 6 > 0\).

Step 3: Solve the Inequality

Now, solve the inequality: \(x + 6 > 0\), by subtracting 6 from both sides. This results in \(x > -6\).

Key Concepts

Domain of a Function

A function's domain is a critical concept in mathematics. It identifies all possible input values, commonly referred to as \(x\)-values, that will provide valid output results.
Understanding the domain helps us avoid calculations that do not make sense, especially with logarithmic functions.

For logarithmic functions like \(f(x) = \log_{b}(g(x))\), it is crucial to only consider values of \(x\) where the argument \(g(x)\) is positive. This is due to the properties of logarithms, which are only defined for positive arguments.

• Logarithm bases need to be positive and not equal to one.
• The argument \(g(x)\) in a logarithmic function must be greater than zero.

In the example of \(f(x) = \log_{5}(x+6)\), we start by focusing on what \(x\) values make the argument \(x+6\) positive, ensuring the function is properly defined. Therefore, finding the domain involves setting \(x + 6 > 0\). This condition tells us about the boundary of \(x\) values that are permissible.

Inequalities

Inequalities are expressions involving the "less than" or "greater than" symbols. They help in understanding relationships between different values.
For logarithmic functions, setting up inequalities ensures the argument of the function remains positive.

Let's consider the inequality \(x + 6 > 0\):

• The inequality symbol \(>\) denotes that \(x+6\) must be greater than zero, ensuring a positive argument.
• Inequalities have different rules from equations, especially when multiplying or dividing by negative numbers.

It's essential to always maintain the correct order, as flipping the inequality symbol incorrectly during calculations is a common mistake.
This is especially true in problems involving functions where the domain constraint naturally leads to inequalities.

Solving Inequalities

Solving inequalities is an important skill necessary to determine the domain of many functions. To solve an inequality like \(x + 6 > 0\), follow these simple steps:

First, isolate \(x\) by performing operations that maintain the inequality's balance:

• Subtract 6 from both sides of the inequality: \(x + 6 - 6 > 0 - 6\).

This simplifies to \(x > -6\), indicating that \(x\) must be greater than -6 to satisfy the inequality.

• Check your result to ensure the solution makes logical sense. Substitute values greater than \(-6\) back into the original expression \(x+6\).
• Any value greater than \(-6\) ensures a positive argument, validating the domain requirement for the logarithm.

By understanding these steps, you can solve inequalities effectively and determine valid input ranges for logarithmic functions.