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Chapter 6: Problem 43

Find the domain of each function. $$ f(x)=3-2 \log _{4}\left(\frac{x}{2}-5\right) $$

Short Answer

Expert verified
The domain is \( x > 10 \).

Step by step solution

01

- Identify the logarithmic function

The given function is \( f(x) = 3 - 2 \, \log_{4} \left( \frac{x}{2} - 5 \right) \). Recognize that the domain of a logarithmic function \( \log_{a}(b) \) requires that the argument \( b \) must be greater than zero.
02

- Set the argument greater than zero

Set up the inequality for the logarithmic argument: \( \frac{x}{2} - 5 > 0 \). This condition must be satisfied for the logarithm to be defined.
03

- Solve the inequality

Solve the inequality \( \frac{x}{2} - 5 > 0 \): 1. Add 5 to both sides: \( \frac{x}{2} > 5 \)2. Multiply both sides by 2: \( x > 10 \).
04

- State the domain

The domain of the function \( f(x) = 3 - 2 \, \log_{4} \left( \frac{x}{2} - 5 \right) \) is all real numbers greater than 10.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Functions
Logarithmic functions are the inverses of exponential functions. For a logarithmic function of the form \( \text{log}_a(b) \), the base \( a \) must be a positive real number, and \( b \), called the argument, must be greater than zero. This is because you can only take logs of positive values in real-number mathematics. So, if you have a function like \( f(x) = 3 - 2 \, \text{log}_4 \big( \frac{x}{2} - 5 \big) \), you need to ensure the argument \( \frac{x}{2} - 5 \) is greater than zero to keep the function valid.
Inequalities
To determine the domain of a logarithmic function, we need to solve an inequality. The inequality comes from the requirement that the argument of the logarithm must be greater than zero. For our function, the inequality is \( \frac{x}{2} - 5 > 0 \). By solving this inequality, we figure out the range of possible \( x \) values that make the function valid.
Function Analysis
To analyze functions, we break them down into components. For the function \( f(x) = 3 - 2 \, \text{log}_4 \big( \frac{x}{2} - 5 \big) \), we note that it's a combination of a logarithmic function, a constant, and a linear scaling factor. The root task here is identifying and solving the component involving the logarithm. This involves ensuring that the argument inside the log (i.e., \( \frac{x}{2} - 5 \)) is valid (greater than zero).
Domain and Range
In mathematics, the domain of a function is the set of all possible input values (or \( x \)-values) that the function can accept without resulting in undefined or non-real outputs. For logarithmic functions, this means finding where the argument is positive. Solving the inequality \( \frac{x}{2} - 5 > 0 \) results in:
  • Add 5 to both sides: \( \frac{x}{2} > 5 \)
  • Multiply both sides by 2: \( x > 10 \)
This gives us the domain \( x > 10 \). The range, generally, for such a logarithmic function depends on the transformations applied and is not limited in this context.

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