Math Problem Statement

Find the equation of the vertical asymptote f(x) = log(x - 3)

Solution

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image_path = '/mnt/data/1733728231432paysi2wz.jpg' img = Image.open(image_path) img.show() # This will display the image so we can analyze it visuallyIt seems that I am unable to display or interpret the image directly. However, I can use text recognition (OCR) to extract any mathematical content or text from the image. Let me process that for you. pytesseract

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textThe image contains the following problem:

Find the equation of the vertical asymptote for the function
f(x)=log(x3)f(x) = \log(x - 3)

Solution:

The function f(x)=log(x3)f(x) = \log(x - 3) is a logarithmic function. A logarithmic function logb(xh)\log_b(x - h) has a vertical asymptote where the argument of the logarithm is equal to zero, i.e., xh=0x - h = 0. This is because the logarithm of zero is undefined, and the function approaches negative infinity as xx approaches the asymptote.

For the given function f(x)=log(x3)f(x) = \log(x - 3), the vertical asymptote occurs where: x3=0x - 3 = 0 So, the vertical asymptote is: x=3x = 3

Answer:

The equation of the vertical asymptote is x=3x = 3.

Would you like more details or have further questions on this topic?

Here are some related questions for you to explore:

  1. How does the vertical asymptote affect the graph of the logarithmic function?
  2. What happens to the function f(x)=log(xh)f(x) = \log(x - h) for different values of hh?
  3. How do you find horizontal asymptotes for other types of functions (e.g., rational functions)?
  4. What is the domain of the function f(x)=log(x3)f(x) = \log(x - 3)?
  5. How do transformations affect the location of asymptotes?

Tip: To identify the vertical asymptote of any logarithmic function f(x)=logb(xh)f(x) = \log_b(x - h), solve the equation xh=0x - h = 0. This gives the value where the asymptote is located.

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Math Problem Analysis

Mathematical Concepts

Logarithmic Functions
Vertical Asymptotes

Formulas

x - h = 0 for vertical asymptote of log(x - h)

Theorems

Domain of logarithmic functions

Suitable Grade Level

Grades 10-12