Problem 78 - College Algebra

Question

Find the domain of each logarithmic function. $$ f(x)=\log (7-x) $$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\log (7-x)\) is \(x < 7\).

Step 1: Writing the condition for the domain

The domain of a logarithmic function is all x-values that make the expression within the logarithm positive. In this case, the expression within the logarithm is \(7-x\). Therefore, \(7-x > 0\) that is, the function is defined for values of x which satisfy this inequality.

Step 2: Solving the inequality

After rearranging the inequality by adding x to both sides and subtracting 0 from both sides, the inequality becomes: \(x < 7\).

Step 3: Writing the domain

After solving the inequality, you get the domain of the function. The domain of the function \(\log (7-x)\) is \(x < 7\). This means that the function is defined for all real numbers less than 7.

Key Concepts

Domain of a Function

The concept of the domain of a function is fundamental in mathematics. It refers to the set of all possible input values (usually represented as "x") for which the function is defined. In simpler terms, it's the collection of all x-values that you can plug into the function without breaking any mathematical rules, like dividing by zero or taking the logarithm of a negative number.
In the context of logarithmic functions, such as in our example \( f(x) = \log(7-x) \), the requirement is that the expression within the logarithm must be positive. This is because logarithms of zero or negative numbers are not defined in the realm of real numbers.
To find the domain:

• Identify the expression inside the logarithm.
• Set up an inequality where this expression is greater than zero.
• Solve the inequality to find the range of permissible x-values.

This process ensures the function remains valid for all x-values within the found domain.

Inequalities

Inequalities are a way to express that quantities are not equal and show the relationship between them through symbols like '<', '>', '≤', and '≥'. In mathematical terms, inequalities state that one side is bigger, smaller, or equal under specific conditions. Solving inequalities involves finding the values of variables that make the inequality true.
For example, in the function \( f(x) = \log(7-x) \), we set up the inequality \( 7-x > 0 \). Solving this:

• Firstly, rearrange the inequality by moving \(x\) to the other side to obtain \(7 > x\) or equivalently \(x < 7\).

Solving inequalities often involves similar steps to solving regular equations:

• Isolate the variable on one side by performing operations like adding, subtracting, multiplying, or dividing both sides.

Unlike equalities, when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. Understanding and solving inequalities is crucial for finding the domain of many functions.

Real Numbers

Real numbers are a broad category of numbers that include all possible numbers along the number line. This includes:

• Whole numbers like 0, 1, 2, 3, etc.
• Integers, which are whole numbers and their negatives, like -1, -2, -3, etc.
• Rational numbers, which are fractions like \( \frac{1}{2}, \frac{3}{4}, -\frac{2}{3}\).
• Irrational numbers like \( \pi \) or \( \sqrt{2} \), which cannot be expressed as exact fractions.

Sometimes, students confuse real numbers with types of numbers that aren't used as frequently, like complex numbers, which include imaginary numbers and aren't on the standard number line.
Understanding real numbers is vital when figuring out the domain of functions. When calculating domains, like in the case of \( f(x) = \log(7-x) \), we focus on real numbers because that's the primary number system where these functions are used. Thus, a function like this one will only be defined where it produces real-number outputs.