Math Problem Statement

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

Solution

The function provided is f(x)=log(x+3)f(x) = \log(x + 3). To find the vertical asymptote of a logarithmic function, recall that the logarithm is undefined for inputs less than or equal to zero. Therefore, the argument x+3>0x + 3 > 0 determines the domain of the function.

Steps:

  1. Solve x+3>0x + 3 > 0: x>3x > -3

  2. The logarithmic function f(x)=log(x+3)f(x) = \log(x + 3) is undefined at x=3x = -3, and as x3+x \to -3^+, f(x)f(x) \to -\infty.

  3. Vertical Asymptote: The vertical asymptote is at: x=3x = -3

Would you like further explanation or related problems?

Related Questions:

  1. How do you determine the vertical asymptote for other logarithmic functions?
  2. What is the domain of f(x)=log(x+3)f(x) = \log(x + 3)?
  3. How would the graph of f(x)=log(x+3)f(x) = \log(x + 3) look, and where is it increasing or decreasing?
  4. How does the shift x+3x + 3 affect the graph of the parent function log(x)\log(x)?
  5. Can a logarithmic function have more than one vertical asymptote?

Tip:

For logarithmic functions f(x)=log(ax+b)f(x) = \log(a \cdot x + b), always check when the argument ax+b0a \cdot x + b \leq 0 to determine restrictions on the domain and the vertical asymptote.

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Math Problem Analysis

Mathematical Concepts

Logarithmic Functions
Vertical Asymptotes
Domain of Functions

Formulas

Vertical asymptote occurs where the argument of the logarithm is zero (x + c = 0)

Theorems

Properties of Logarithms

Suitable Grade Level

Grades 10-12