Problem 66 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The domain of the function \(g(x)=\log _{4} x\) is \((0, \infty)\). The functions vertical asymptote is \(x = 0\) and the x-intercept is \(x = 1\). The graph increases from left to right, passing through the point \(x = 1\), with a vertical asymptote at \(x = 0\).

Step 1: Find the Domain

The domain of a logarithmic function is always \((0, \infty)\), because you cannot take the logarithm of a negative number or zero. Therefore, the domain of the function \(g(x) = \log _{4} x\) is also \((0, \infty)\)

Step 2: Identify the Vertical Asymptote

The vertical asymptote of a logarithmic function is always \(x = 0\), because the function is undefined at \(x = 0\). So, the vertical asymptote of the function \(g(x) = \log _{4} x\) is \(x = 0\).

Step 3: Find the x-intercept

To find the x-intercept, set the function equal to zero and solve for \(x\). \[\log _{4} x=0\] This implies \(x = 1\), because any number raised to the power of 0 is 1. Therefore, the x-intercept of the function \(g(x) = \log _{4} x\) is \(x = 1\)

Step 4: Sketch the Graph

With the obtained information, draw the graph. Mark the vertical asymptote at \(x = 0\) . The graph will approach but never touch this line. Next, mark the x-intercept at \(x = 1\). The graph will pass through this point. The graph rises from negative infinity along the asymptote and passes through the x-intercept, then continues to rise to infinity.

Key Concepts

Domain of a Logarithmic Function

Understanding the domain of a logarithmic function is crucial when we’re dealing with these types of equations. Simply put, the domain is the set of all possible input values (x-values) for which the function is defined. For the function \(g(x) = \log_{4}x\), the domain consists of all positive real numbers—more technically, \(0, \infty\). This is because the logarithm of zero or any negative number is undefined, as it does not produce a real number result.

Think of a logarithmic function as asking the question, 'To what power do we raise the base to get x?'. For example, with \(g(x)\), we are finding to what power we must raise 4 to get the x-value. Since raising a number to any power cannot result in a negative or zero, the function isn’t defined for these values—hence, they are not included in the domain.

Vertical Asymptote

A vertical asymptote is a vertical line that the graph of a function approaches but never actually touches. In the case of the logarithmic function \(g(x) = \log_{4}x\), this vertical asymptote occurs at \(x = 0\).

Why is this the case? As x approaches zero from the right side, the output of \(g(x)\) decreases without bound, never settling at a finite value. So, the graph eternally approaches but never crosses or touches the y-axis. This behaviour of a graph around a vertical asymptote is a hallmark of logarithmic functions and is important when sketching or recognizing their graphs.

X-Intercept

The x-intercept of a function is the point where the graph crosses the x-axis. To find this point for the function \(g(x) = \log_{4}x\), we set \(g(x)\) equal to zero and solve for x: \[\log_{4}x = 0\], which simplifies to \(x = 1\). That’s because any nonzero number raised to the power of zero equals one.

On the graph, this corresponds to the lone spot where the curve intersects the x-axis. It’s the only point where the height of the graph (the y-value) is zero, intuitively showing us the base-case scenario where the power we’re raising 4 to in order to get x is zero. Therefore, the x-intercept is a fundamental point used in plotting the graph of a logarithmic function.

Graphing Logarithms

Graphing logarithmic functions like \(g(x) = \log_{4}x\) involves a few specific steps. First, acknowledge the domain—remember, only positive x-values are included. Then, identify the vertical asymptote; for \(g(x)\), it’s the y-axis (\(x = 0\)).

Next, find the x-intercept. As we discussed earlier, for \(g(x)\), this is at \(x = 1\). Plot this point on the graph, as it provides a key reference for the curve. When sketching, start drawing the curve just to the right of the vertical asymptote; it should approach this asymptote but never touch it. As you draw away from the y-axis and pass through the x-intercept, the curve will continue to rise gently, getting closer to horizontal as x increases.

Remembering this general shape—and that the function's value increases as x increases—will help you sketch an accurate logarithmic graph.