Problem 68 - College Algebra and Calculus: An Applied Approach

Question

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. Then sketch its graph.\(f(x)=\log _{4}(x-3)\)

Step-by-Step Solution

Answer

The domain of the logarithmic function is \(x > 3\), the vertical asymptote is \(x = 3\), and the x-intercept is \(x = 4\). And the graph increases when \(x > 3\).

Step 1: Find the Domain of the Function

The domain of a logarithmic function is determined by the argument of the logarithm. In this case the argument is \(x-3\). The argument must be greater than zero, so the domain is \(x > 3\).

Step 2: Find the Vertical Asymptote

The vertical asymptote of a logarithmic function can be found by solving the inequality of the argument to be less than or equal to zero. In this case, the equation is \(x-3 ≤ 0\), which gives \(x=3\). Therefore, the vertical asymptote is \(x=3\).

Step 3: Find the x-intercept

The x-intercept is found by setting \(f(x)\) to zero and solving for \(x\). Thus, \(0= \log_{4}(x-3)\). We use the definition of logarithms to rewrite as \(4^0 = x - 3\). Therefore, it solves as \(x=4\), so the x-intercept is \(x=4\).

Step 4: Sketch the Graph

Now to sketch the graph, use the characteristics of the function found: The domain, vertical asymptote \(x=3\), and x-intercept \(x=4\). Plot these on the graph to visualize and plot additional points if needed for clarity. The graph increases as \(x\) increases when \(x>3\).

Key Concepts

Domain of a function

The domain of a function refers to all the possible values that can be used as input to the function. For a logarithmic function like \( f(x) = \log_4(x-3) \), the argument inside the logarithm determines the domain.
In this case, the argument is \( x - 3 \). Since logarithms are only defined for positive numbers, we set \( x - 3 > 0 \). Solving this inequality gives us \( x > 3 \). This means the domain of the function is all real numbers greater than 3.

• The expression \( x - 3 \) must be positive.
• Thus, the valid set of \( x \) values is \( x > 3 \).
• The domain can be expressed in interval notation as \( (3, \infty) \).

Vertical asymptotes

Vertical asymptotes occur in functions where at certain \( x \) values, the function grows toward positive or negative infinity. For logarithmic functions, vertical asymptotes are found where the argument inside the logarithm is zero.
In \( f(x) = \log_4(x - 3) \), the vertical asymptote is found by solving the equation \( x - 3 = 0 \). This gives \( x = 3 \), meaning there is a vertical asymptote at \( x = 3 \).

• Vertical asymptotes mark boundaries where the function is undefined.
• They can be visualized as a line on a graph that the curve approaches but never touches.
• For \( \log(x - 3) \), if \( x \) approaches 3 from the right, \( f(x) \) heads to negative infinity.

X-intercepts

The x-intercept of a graph is the point where the function crosses the x-axis. For a function like \( f(x) = \log_4(x-3) \), the x-intercept is found by setting the function equal to zero and solving for \( x \).
Setting \( f(x) \) to zero gives \( 0 = \log_4(x - 3) \). To solve this, translate the logarithmic equation to its exponential form: \( 4^0 = x - 3 \), which simplifies to \( 1 = x - 3 \). Solving for \( x \), we find \( x = 4 \). Therefore, the x-intercept is at \( x = 4 \).

• Set the function output \( f(x) = 0 \) to find the x-intercept.
• The equation simplifies using logarithm properties \( \log_b(b^y) = y \).
• The intercept represents a specific point on the x-axis (here, \( x = 4 \)).