Problem 64 - Precalculus Mathematics for Calculus

Question

Find the domain of the function. $$f(x)=\log _{5}(8-2 x)$$

Step-by-Step Solution

Answer

The domain is \(x < 4\) or \((-\infty, 4)\).

Step 1: Identify the Condition for the Logarithm

A logarithmic function is defined only when its argument is positive. Therefore, for the function \(f(x) = \log_{5}(8-2x)\), the argument \(8-2x\) must be greater than zero. This can be expressed as the inequality \(8-2x > 0\).

Step 2: Solve the Inequality

We need to solve the inequality \(8-2x > 0\). Begin by isolating \(x\) on one side. Start by subtracting 8 from both sides: \(-2x > -8\). Then, divide both sides by \(-2\), remembering to reverse the inequality sign: \(x < 4\).

Step 3: Write the Domain Based on the Solution

The solution of the inequality \(x < 4\) determines the domain of the function. Therefore, the domain of \(f(x) = \log_{5}(8-2x)\) is all real numbers \(x\) such that \(x < 4\). In interval notation, this is expressed as \((-\infty, 4)\).

Key Concepts

Logarithmic Function

Logarithmic functions are a fundamental part of mathematics, especially when it comes to dealing with exponential relationships. A logarithm answers the question: "To what exponent must we raise a certain base to obtain a given number?" For example, in the function \( \log_{5}(8-2x) \), the base is 5. We are trying to find what power we must raise 5 to get the result of \( 8-2x \).
Logarithmic functions are only defined for positive argument values. This means that whatever is inside the logarithm must be more than zero. Why is that? It's because you can only take logs of positive numbers in the real number system. So in our exercise, we set the requirement that \( 8-2x > 0 \). Only these values of \( x \) will keep our expression valid.
Whenever you're faced with a logarithmic function, remember:

• The base of the log is a positive real number different from 1.
• The argument must always be greater than zero.

Understanding these principles allows us to crack the conditions of the logarithmic function effectively.

Inequality

Inequalities are essentially mathematical expressions involving the symbols \( >, <, \leq, \, \geq \). They enable us to express conditions where something isn't necessarily equal but rather lesser or greater. In the context of finding domains, inequalities are indispensable.
Let's revisit our example problem, \( 8-2x > 0 \). What this inequality means is that whatever value \( x \) takes should make \( 8-2x \) a positive number. Solving this inequality involves a couple of strategic steps to isolate \( x \):

• Subtract 8 from both sides, yielding \( -2x > -8 \).
• Divide both sides by -2, making sure to flip the inequality symbol, giving \( x < 4 \).

By reversing the inequality sign when dividing by a negative number, you've maintained the valid condition of the inequality. The solution here, \( x < 4 \), signifies the 'safe zone' of values \( x \) can take to keep our function valid.

Interval Notation

Interval notation provides a convenient and compact way to express a range of numbers, especially when describing domain or solution sets. It's a concise manner of showing which parts of the number line are included and which are not.
In our case with \( x < 4 \), the domain is all real numbers that are less than 4. To express this in interval notation, we write \( (-\infty, 4) \). This tells us:

• The numbers start from negative infinity (very far to the left) and go up until just before 4.
• The use of a parenthesis "(" means 4 is not included; it only approaches this point.

Interval notation gives a crisp and clear picture of all potential solutions or elements in a set. It's particularly helpful when dealing with continuous sets of numbers, like in this logarithmic domain example.