How to find the vertical asymptote of a logarithmic function?
Understand the Problem
The question is asking how to determine the vertical asymptote of a logarithmic function, which typically involves analyzing the function's domain and identifying values that make the argument of the logarithm equal to zero.
Answer
The vertical asymptote is $x = 3$.
Answer for screen readers
The vertical asymptote of the logarithmic function $f(x) = \log_b(x - 3)$ is $x = 3$.
Steps to Solve
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Identify the logarithmic function First, write down the logarithmic function you need to analyze. For example, let’s consider the function $f(x) = \log_b(g(x))$, where $g(x)$ is the argument of the logarithm.
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Determine the argument of the logarithm Identify the expression $g(x)$ inside the logarithm. For example, if our function is $f(x) = \log_b(x - 3)$, then $g(x) = x - 3$.
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Set the argument to zero Set the argument $g(x)$ equal to zero to find the value of $x$ that will make the logarithm undefined. This is crucial because logarithms are undefined for non-positive values. For the example, we solve: $$ x - 3 = 0 $$
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Solve for the value of x Now solve the equation from the previous step. In our example: $$ x - 3 = 0 \implies x = 3 $$
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Identify the vertical asymptote The vertical asymptote of the logarithmic function occurs at the value of $x$ we just found. Thus, the vertical asymptote is $x = 3$.
The vertical asymptote of the logarithmic function $f(x) = \log_b(x - 3)$ is $x = 3$.
More Information
In general, vertical asymptotes for logarithmic functions occur at the values where the argument of the logarithm equals zero. Logarithmic functions have a unique behavior where they approach negative infinity as they approach these points.
Tips
- Forgetting to set the argument equal to zero: Students sometimes skip this crucial step and look for other values mistakenly.
- Confusing the logarithm's base: Ensure you note the base of the logarithm when analyzing vertical asymptotes.
