Problem 70 - College Algebra and Calculus: An Applied Approach

Question

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. Then sketch its graph.\(h(x)=-\log _{4}(x-1)\)

Step-by-Step Solution

Answer

The function \(h(x)= -\log_{4}(x-1)\) has a domain of \((1, \infty)\), a vertical asymptote at \(x=1\), and an x-intercept at \(x=2\). The graph of the function reflects over the x-axis, compared to the basic logarithmic function graph due to the '-log', and approaches but never touches the asymptote at \(x=1\).

Step 1: Determine the Domain

The domain of a logarithmic function is always \((a, \infty)\) where \(a\) is a number which makes the argument of the log function become greater than zero. In this case, to make \(x-1>0\), \(x\) should be greater than \(1\), so the domain of \(h(x)= -\log_4(x-1)\) is \((1, \infty)\).

Step 2: Find the Vertical Asymptote

The vertical asymptote in a logarithmic function is always at the value which makes the argument of the log function zero. In this case, when \(x = 1\), \((x-1)\) becomes \(0\), so there is a vertical asymptote at \(x=1\).

Step 3: Determine the \(x\)-Intercept

To find the x-intercept, set \(h(x)\) to \(0\) and solve for \(x\). Thus, \(-\log_{4}(x-1)=0\). We rewrite this as \(4^0 = x - 1\). Solving for \(x\), we get \(x = 2\). Therefore, the x-intercept is \(x=2\).

Step 4: Sketch the Graph

Now that we have all the necessary points and lines for our graph, draw a vertical asymptote at \(x = 1\). Plot the point at \(x = 2\) (intercept). Remember that the curve approaches but never touches the asymptote. Due to the '-log' the graph will reflect over the x-axis, compared to the basic logarithmic function graph, giving us our final image.

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial in determining the set of possible input values that the function can accept. For logarithmic functions, the domain is specifically defined by the condition that the argument (the expression inside the logarithm) must be greater than zero.
For example, looking at the function given by the exercise:

• The logarithmic function is defined as: \( h(x) = -\log_4(x - 1) \).
• The argument here is \( x - 1 \), meaning we must have \( x - 1 > 0 \) for any real value.
• This implies that \( x > 1 \), making the domain \((1, \infty)\).

The domain tells us that we can use any number greater than 1 as an input to our function, while numbers less than or equal to 1 are not part of the domain, as they would make the argument zero or negative.

Vertical Asymptotes

Vertical asymptotes are lines the graph of a function approaches but never actually touches or crosses. For logarithmic functions, vertical asymptotes occur at the points that make the argument zero.
Let's consider the example from our exercise:

• The function is \( h(x) = -\log_4(x - 1) \).
• Here, the argument \( x - 1 \) becomes zero when \( x = 1 \).
• This tells us that the line \( x = 1 \) is a vertical asymptote.

The existence of a vertical asymptote at \( x = 1 \) means that as the input \( x \) gets closer and closer to 1 (from the right), the value of the function decreases without bound. This behavior is a typical characteristic of how logarithmic functions react to their vertical asymptotes.

x-intercepts

Finding the \( x \)-intercept of a function involves determining the point(s) where the graph of the function crosses the x-axis. At these points, the function's output, or \( y \)-value, is zero.
In the exercise, to find the \( x \)-intercept of \( h(x) = -\log_4(x - 1) \):

• Set the function equal to zero: \(-\log_4(x - 1) = 0\).
• Solve for \( x \): Rewrite this as \( \log_4(x - 1) = 0 \). This implies that \( x - 1 = 4^0 = 1 \).
• Thus, \( x = 1 + 1 = 2 \).

Hence, the \( x \)-intercept is at \( x = 2 \), meaning the graph of the function crosses the x-axis at this point. The \( x \)-intercept provides a tangible point on the graph that confirms where the function reaches zero, which is particularly useful in sketching and understanding the overall behavior of the logarithmic function.