Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the domain of \(y=\log _{4}\left(\frac{x-4}{x}\right)\).

Short answer

The domain is \(-\text{inf} < x < 0 \) \cup\ \( 4 < x < \text{inf}\).

Step-by-step solution

- Recall the properties of logarithms

To find the domain of a logarithmic function, remember that the argument (the value inside the logarithm) must be positive. For this function, the argument is \(\frac{x-4}{x}\) which must be greater than 0.

- Set up the inequality

Write the inequality to find where the argument of the logarithm is positive. \(\frac{x-4}{x} > 0\).

- Solve the inequality

To solve the inequality \(\frac{x-4}{x} > 0\), find the values of x that make the numerator and the denominator zero. The numerator is zero when x = 4, and the denominator is zero when x = 0. This gives us two intervals to consider: \(-\text{inf} < x < 0\) and \(0 < x < 4\).

- Test the intervals

Choose a test point from each interval to check where the inequality holds true. For \(-\text{inf} < x < 0\), try x = -1: \(\frac{-1 - 4}{-1} = \frac{-5}{-1} = 5 > 0\). For \(0 < x < 4\), try x = 1: \(\frac{1 - 4}{1} = \frac{-3}{1} = -3 < 0\). Hence, the inequality holds in the interval \(-\text{inf} < x < 0\).

- Combine with critical points

Since \(\frac{x-4}{x}\) cannot be zero and the logarithm's argument must be positive, exclude the points x = 4 and x = 0 from the domain. Thus, the solution is \(-\text{inf} < x < 0\) in union with \(4 < x < \text{inf}\).

Key concepts

logarithmic functions

Logarithmic functions are mathematical functions that are the inverse of exponential functions. They are critical in various fields such as science, engineering, and finance. A logarithmic function can be represented as:
\(y = \log_b(x)\), where \(b\) is the base of the logarithm, and \(x\) is the argument. The logarithm tells us the power to which the base must be raised to produce the argument. For example, \(\log_2(8) = 3\) because 2 raised to the power of 3 equals 8.
One essential property of logarithms is that their argument must be positive. This is because you cannot take the logarithm of zero or a negative number in the set of real numbers.

inequality solving

Solving inequalities is a crucial part of determining the domain of logarithmic functions. Inequalities indicate where a function's argument is valid. For example, to find the domain of \(y = \log_4\left(\frac{x-4}{x}\right)\), you need to solve the inequality \(\frac{x-4}{x} > 0\).
When solving this kind of inequality, break it down into steps:

• Identify where the numerator and denominator of the fraction become zero.
• Set up intervals based on these critical points.
• Choose test points in these intervals to determine where the inequality holds true.

In our example, the numerator zeroes out at \x = 4\, and the denominator zeroes out at \x = 0\. This gives us the intervals to test: \{-\inf < x < 0\}\ and \{0 < x < 4\}. By testing values within these intervals, you identify where the function argument remains positive.

function domain

The domain of a function defines all the possible input values (x-values) for which the function is defined. For logarithmic functions, determining the domain often requires solving inequalities to ensure the argument inside the logarithm is always positive. Let's break down the domain determination for \(y = \log_4\left(\frac{x-4}{x}\right)\).
First, set up the inequality \(\frac{x-4}{x} > 0\) to find where the argument is positive. Solving this inequality involves:

• Finding critical points (where numerator and denominator are zero)
• Setting up and testing intervals
• Combining intervals where the inequality holds

In this function, \x = 4\ and \x = 0\ are the critical points. Testing intervals gives us \{-\inf < x < 0\}\ and \{4 < x < inf\}. Thus, the domain is \{-\inf < x < 0\}\ union \{4 < x < inf\}. Remember, the argument of the logarithmic function must never be zero or negative to keep the function valid.