Problem 59 - Precalculus

Question

State the domain of the logarithmic function in interval notation. $$f(x)=\ln (7-2 x)$$

Step-by-Step Solution

Answer

The domain of the function is \((-\infty, \frac{7}{2})\).

Step 1: Understanding the Domain of Logarithmic Functions

The domain of a logarithmic function is determined by the requirement that the argument (the expression inside the logarithm) must be greater than zero. This is because the logarithm of zero or a negative number is undefined.

Step 2: Setting up the Inequality

For the function \( f(x) = \ln(7 - 2x) \), the argument \( 7 - 2x \) needs to be greater than zero. Therefore, the inequality we need to solve is:\[7 - 2x > 0\]

Step 3: Solving the Inequality

Rearrange the inequality \( 7 - 2x > 0 \) to isolate \( x \):1. Subtract 7 from both sides: \(-2x > -7\)2. Divide both sides by -2 (remember to flip the inequality sign when dividing by a negative number): \[x < \frac{7}{2}\]Thus, \( x \) must be less than \( \frac{7}{2} \).

Step 4: Finding the Domain in Interval Notation

The solution to the inequality \( x < \frac{7}{2} \) gives us all \( x \) values that are less than \( \frac{7}{2} \). Since \( x \) can be any real number less than \( \frac{7}{2} \), the domain of the function in interval notation is:\[(-\infty, \frac{7}{2})\]

Key Concepts

Properties of Logarithms

Logarithms are essential mathematical functions that help in solving equations related to exponential growth and decay. Understanding the properties of logarithms can simplify complex expressions and inequalities. Here are some important properties that come in handy:

• Product Property: \[\log_b (xy) = \log_b x + \log_b y\]This property states that the logarithm of a product is equal to the sum of the logarithms of the factors.
• Quotient Property:\[\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y\]It represents the logarithm of a quotient as the difference of the logarithms.
• Power Property:\[\log_b (x^n) = n \log_b x\]This indicates that the logarithm of a power is the exponent multiplied by the logarithm of the base.

These properties make it easier to solve logarithmic equations and inequalities, transforming them into more manageable forms. Always remember, the base of the logarithm should be positive and not equal to 1 to ensure these properties operate correctly.

Solving Inequalities

Inequalities involve finding the range of values that satisfy an expression instead of a single solution. In our exercise, we needed to solve an inequality:\[7 - 2x > 0\]Inequalities require a few extra rules compared to equations:

• When multiplying or dividing by a negative number, reverse the inequality sign. For instance, dividing by \(-2\) changes \(>\) to \(<\).
• Keep the variable on one side and constants on the other using addition or subtraction.

Let's see how we applied these rules:
1. Subtract 7 from both sides: \[-2x > -7\]
2. Divide by \(-2\) and reverse the inequality sign: \[x < \frac{7}{2}\]
This approach leads to finding all values of \(x\) that satisfy the expression, paving the way towards understanding function domains.

Interval Notation

Interval notation offers a convenient method to express the set of solutions of inequalities or to describe the domain and range of functions. For the exercise given, the function domain was expressed using interval notation.

• Square brackets \([,]\) are used when a number is included in the interval.
• Parentheses \((,)\) are used to exclude a number, showing it doesn't belong to the interval.
• "-\(\infty\)" and "\(\infty\)" always use parentheses because infinity is a concept, not a number you can reach or include.

For example, \((-\infty, \frac{7}{2})\) denotes all real numbers less than \(\frac{7}{2}\), excluding \(\frac{7}{2}\) itself. Interval notation provides a clear, concise representation of sets, useful in expressing solutions and domains.