Problem 69 - College Algebra and Calculus: An Applied Approach

Question

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. Then sketch its graph.\(f(x)=-\log _{2} x\)

Step-by-Step Solution

Answer

The domain of the function \(f(x)=-\log _{2} x\) is \(x > 0\). The vertical asymptote is at \(x = 0\). The \(x\)-intercept is at \(x=1\). The function's graph starts at the point (1,0) in the 1st quadrant and slopes downwards to the right, which continues in the negative y-axis of the 4th quadrant.

Step 1: Find the Domain

The domain of a function represents all the values that \(x\) can take on. In a logarithmic function \(f(x) = \log_{a}x\), the base \(a\) is always positive and not equal to 1, and \(x\) must always be greater than 0. Therefore, the domain of \(f(x)=-\log _{2} x\) is \(x > 0\).

Step 2: Find the Vertical Asymptote

The vertical asymptote of a logarithmic function \(f(x) = \log_{a}x\) is a vertical line at \(x = 0\). This is because as \(x\) approaches zero, the logarithmic function approaches negative infinity.

Step 3: Finding the \(x\)-intercept

The \(x\)-intercept is the value of the \(x\) at which the function \(f(x)\) equals to zero. Solving \(-\log _{2} x = 0\), we get \(\log _{2} x = 0\). Therefore, the x-intercept is \(x=1\) using the property of logarithms that \(log_{a}1 = 0\) for any positive number \(a\) not equal to 1.

Step 4: Sketch the Graph

Sketching the graph requires plotting the points on a graph and drawing the curve. The graph will start at the point (1,0) and decrease downwards to the right. It will follow a curve that approaches but never reaches the y-axis, hence showing the asymptote at \(x=0\). The graph will be in the 1st and 4th quadrant due to the property of the logarithmic function including negative values.

Key Concepts

Domain of a Logarithmic Function

Understanding the domain of a logarithmic function is essential for analyzing its behavior. In simple terms, the domain includes all the possible values of 'x' you can plug into the function without causing an error in the computation.

For the function in question, \(f(x) = -\log_2(x)\), the domain is defined by the condition that 'x' must be positive, meaning \(x > 0\). This is because logarithms of zero or negative numbers are not defined within real numbers. The fact that 'x' cannot be zero or negative is an inherent property of logarithms, as they answer the question: 'To what power must the base (2 in this case) be raised, to obtain 'x'?' This question only makes sense for values of 'x' that are positive.

Why can't 'x' be negative or zero? To understand this, consider the fact that raising any positive base to any power will never result in a negative number or zero. Hence, the set of all positive real numbers is the domain for our function \(f(x) = -\log_2(x)\).

Vertical Asymptote of Logarithmic Functions

A vertical asymptote is a line that the graph of a function approaches but never touches or crosses. For logarithmic functions, such as \(f(x) = -\log_2(x)\), the vertical asymptote is located at \(x = 0\).

But why does the logarithmic function have a vertical asymptote at \(x = 0\)? It's because as 'x' gets closer and closer to zero from the positive side, the value of \(\log_2(x)\) decreases without bound—or in other words, it goes off to negative infinity. Since the function can never take the value of zero (because the log of zero is undefined), the graph will infinitely approach, but never actually reach the y-axis. This creates a vertical boundary on the graph of \(f(x)\text{, which is the vertical asymptote at }x = 0\).

X-Intercept of Logarithmic Functions

The x-intercept of a function is a fundamental aspect of its graph, representing the point where the function crosses the x-axis. In terms of coordinates, it's where the function has an output value of zero (\(f(x) = 0\)).

For the logarithmic function \(f(x) = -\log_2(x)\), we find the x-intercept by setting the function equal to zero and solving for 'x'. This gives us \(-\log_2(x) = 0\), which simplifies to \(\log_2(x) = 0\) when we divide both sides by negative one. By understanding the properties of logarithms, we know that the only solution to \(\log_2(x) = 0\) is when \(x = 1\), because any base raised to the power of zero is equal to one.

Therefore, the x-intercept is the point \((1, 0)\) on the graph of our function, which is where the function crosses the x-axis.

Sketching Logarithmic Graphs

Sketching the graph of a logarithmic function, like \(f(x) = -\log_2(x)\), requires an understanding of its behavior and characteristics, such as the domain, x-intercept, and vertical asymptote.

To sketch the graph, start by marking the x-intercept, which we found to be at \((1, 0)\). Next, draw the vertical asymptote at \(x = 0\) to remind us that the graph will never touch this line. Since \(f(x)\) is negative, we know that our graph will be a reflection of the typical log graph across the x-axis. This means that instead of increasing, the graph decreases as 'x' moves away from the asymptote.

Plot additional points if necessary to get a sense of the curvature. The graph will descend into the fourth quadrant as 'x' increases, illustrating that as our input grows, the output (while negative) becomes less negative. Finally, connect the points to show that the graph gradually flattens out as 'x' increases while steeply dropping as 'x' approaches zero—yet never actually reaching the vertical asymptote.