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Chapter 13: Problem 71

Explain how to graph a logarithmic function of the form \(f(x)=\log _{a} x\)

Short Answer

Expert verified
To graph a logarithmic function of the form \(f(x) = \log_a{x}\), follow these steps: 1. Identify the base 'a' and the parent function \(y = \log_{a}{x}\). 2. Understand the properties of logarithmic functions, including domain, range, and existence of a vertical asymptote at \(x = 0\). 3. Determine the graph's behavior depending on whether the base 'a' is greater than 1 or between 0 and 1. 4. Identify key points on the graph, such as the point (1, 0) and the intersection with the x-axis. 5. Sketch the graph using the information from previous steps, including the vertical asymptote, key points, and behavior of the graph.

Step by step solution

01

Identify the base and parent function

Identify the base 'a' of the logarithmic function given as \(f(x)=\log _{a} x\). The parent function for this equation is \(y = \log_{a}{x}\).
02

Know the properties of logarithmic functions

Understand the basic properties of logarithmic functions, which will help you determine the shape, domain, range, and asymptote of the graph. Some of the key properties include: 1. Domain: The valid values for x are all positive numbers: \((0, +\infty)\) 2. Range: The values for y will span all real numbers: \((-\infty, +\infty)\) 3. The function will always have a vertical asymptote at \(x = 0\). 4. The graph of the function will pass through the point (1, 0), since \(\log_a(1) = 0\).
03

Determine the graph's behavior

Determine whether the base 'a' is greater than 1 or between 0 and 1. This will affect the behavior of the graph. Case 1: If \(a > 1\), the graph will increase as you move right (x increases). Example: \(y=\log_{2}{x}\) Case 2: If \(0 < a < 1\), the graph will decrease as you move right (x increases). Example: \(y=\log_{\frac{1}{2}}{x}\)
04

Identify key points

Identify the key points on the graph that will help you sketch the graph accurately. Some key points include (1, 0) and the point at which the graph intersects the x-axis (if it exists). To find the intersection with the x-axis, solve for x when \(f(x) = 0\): \(\log_a x = 0\) Applying the definition of logarithms: \(a^0 = x\) This implies \(x = 1\), so the graph intersects the x-axis at the point (1, 0).
05

Sketch the graph

Use the information gathered in previous steps to sketch the graph of the function: 1. Draw the vertical asymptote at x = 0. 2. Mark the point (1, 0) on the graph. 3. Determine whether the graph increases or decreases as you move right (based on the base 'a'). 4. Sketch the curve of the logarithmic function based on the above factors, ensuring that the graph smoothly approaches the vertical asymptote and passes through the key points. By following these steps, you will have successfully graphed a logarithmic function of the form \(f(x) = \log_a x\).

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Most popular questions from this chapter

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