Problem 56 - Precalculus

Question

State the domain of the logarithmic function in interval notation. $$f(x)=\log _{2}(4 x-1)$$

Step-by-Step Solution

Answer

The domain of the function is \((\frac{1}{4}, \infty)\).

Step 1: Understand the Logarithm Function's Domain

The domain of a logarithmic function \(f(x) = \log_b(g(x))\) is determined by the condition that the argument \(g(x)\) must be greater than zero. This means we need \(g(x) > 0\).

Step 2: Set Up the Inequality

For the given function, \(f(x) = \log_2(4x-1)\), the argument is \(4x-1\). Set up the inequality: \(4x-1 > 0\).

Step 3: Solve the Inequality

Solve \(4x-1 > 0\) by first adding 1 to both sides to get \(4x > 1\). Then, divide each side by 4 to obtain \(x > \frac{1}{4}\).

Step 4: Express the Solution in Interval Notation

Convert the inequality solution \(x > \frac{1}{4}\) into interval notation. The domain in interval notation is \((\frac{1}{4}, \infty)\).

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values, or "x" values, that will give an output in the function. For logarithmic functions, like any other, identifying the domain is essential.
When it comes to logarithmic functions specifically, the argument of the logarithm (what’s inside the log) must always be positive. This is because the logarithm of zero or a negative number is undefined in the real number system.
To determine the domain of a function like \( f(x) = \log_{2}(4x-1) \), we first look at the expression that follows the logarithm: \( 4x - 1 \). The condition for the domain here becomes \( 4x - 1 > 0 \). This sets the requirement that any input \( x \), when plugged into \( 4x - 1 \), must result in a value greater than zero. This is crucial as it ensures the logarithm is defined.

Inequalities

Inequalities are mathematical expressions involving symbols such as \( >, <, \geq, \leq \) that show the relative size or order of two values. They play an important role when finding domains of functions.
In the context of our function \( f(x) = \log_{2}(4x-1) \), we use the inequality \( 4x - 1 > 0 \) to find the domain. Solving this inequality involves simple algebraic steps:

• First, you add 1 to both sides of the inequality: \( 4x - 1 + 1 > 0 + 1 \) which simplifies to \( 4x > 1 \).
• Next, divide each side of the inequality by 4 to solve for \( x \): \( x > \frac{1}{4} \).

Understanding inequalities helps in determining what values are permissible for \( x \) so that the function is valid and makes sense mathematically.

Interval Notation

Interval notation is a way of writing subsets of the real number line. It describes a range of values that \( x \) can take. This is often the result once an inequality is solved.
For a solution like \( x > \frac{1}{4} \), interval notation is used to express this range efficiently. When you see \((\frac{1}{4}, \infty)\), it describes all numbers greater than \( \frac{1}{4} \), up to infinity.

• The parentheses "( )" are used because \( x = \frac{1}{4} \) is not included (as opposed to "[ ]" which would mean inclusive).
• The infinity symbol \( \infty \) always has a parenthesis, as we can never exactly "reach" infinity.

Interval notation is a concise way to represent solutions to inequalities and helps clearly convey the domain of functions like logarithms.