Problem 101 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the logarithmic function. Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. $$f(x)=\log _{3}(x-4)$$

Step-by-Step Solution

Answer

The plot of the function \(f(x)= \log_{3} (x-4)\) shows it is shifted 4 units to the right of the base function. The domain of the function is \( (4, \infty) \). The function has a vertical asymptote at \(x=4\), and it intersects the x-axis at \(x=5\).

Step 1: Plotting function

First, use a graphing utility (like a graphing calculator or software) to plot the function \(f(x)= \log _{3}(x-4)\) and visualize its course. Based on properties of logarithms, this is a transformation of the base function \( \log_{3} x \), shifted 4 units to the right.

Step 2: Identifying domain

Next, identify the domain of the function. The domain of a logarithmic function is all the values that \(x\) can take for which the function is defined. In the case of \(f(x)= \log _{3}(x-4)\), the function is defined for all \(x>4\). Therefore, the domain is \( (4,\infty) \).

Step 3: Identifying vertical asymptote

Now, find the vertical asymptote of the function. The vertical asymptote of the logarithmic function \(f(x)= \log _{3}(x-4)\) is the vertical line at the left limit of its domain. So here, the vertical asymptote is \(x=4\).

Step 4: Finding x-intercept

Lastly, determine the x-intercept of the function. The x-intercept is the value of \(x\) for which \(f(x)=0\). Setting \(f(x)= \log _{3}(x-4) = 0\), and solving for \(x\), gives \(x=5\). Therefore, the x-intercept is \(x=5\).

Key Concepts

Understanding the Domain of Logarithmic Functions

The domain of a logarithmic function includes all the x-values for which the function is defined. Logarithmic functions like \(f(x) = \log_{3}(x-4)\) have a fundamental constraint: the argument of the logarithm must be greater than zero. This is due to the fact that the logarithm of a non-positive number is undefined.
To determine the domain for \(f(x) = \log_{3}(x-4)\), set the argument greater than zero:

• \(x - 4 > 0\)
• Hence, \(x > 4\)

This inequality tells us that the domain of the function is \((4, \infty)\). Any number greater than 4 can be plugged into the function, while values less than or equal to 4 cannot.

Vertical Asymptote in Logarithmic Functions

A vertical asymptote is a vertical line that a graph approaches but never touches or crosses. In the function \(f(x) = \log_{3}(x-4)\), this asymptote represents the boundary of the domain. This is where the function becomes undefined because the argument inside the logarithm equals zero.
For our function, set the argument equal to zero to find the asymptote:

• \(x - 4 = 0\)
• This solves to \(x = 4\)

So, the vertical asymptote for \(f(x) = \log_{3}(x-4)\) is the line \(x = 4\). At \(x = 4\), the function is undefined, and the graph shoots upwards towards positive infinity or downwards towards negative infinity as it approaches this line.
This occurs because the logarithm's argument is approaching zero, which logarithmically translates to spanning from negative infinity to positive infinity in terms of function height.

Finding the X-Intercept of Logarithmic Functions

The x-intercept of a function is the point where the graph crosses the x-axis, which means the output \(f(x)\) is zero at this point. For the function \(f(x) = \log_{3}(x-4)\), set the function equal to zero to determine where it intercepts the x-axis:

• \(\log_{3}(x-4) = 0\)

This can be rewritten as:

• \(x - 4 = 3^0\)
• Since \(3^0 = 1\), then \(x - 4 = 1\)
• Adding 4 to both sides of the equation gives \(x = 5\)

Thus, the x-intercept is at \((5, 0)\). At this point on the graph, \(f(x)\) crosses the x-axis, indicating this is where the function has a zero output, or a value of y equal to zero.