How to find asymptotes of a logarithmic function?
Understand the Problem
The question is asking for the method to determine the asymptotes of a logarithmic function. Asymptotes are lines that the graph of a function approaches but never touches. In the case of logarithmic functions, it's common to find vertical asymptotes, which occur where the function is undefined, typically at points where its argument is zero or negative.
Answer
The vertical asymptote is at $x = -\frac{c}{b}$ for the function $y = \log_a(bx + c)$.
Answer for screen readers
The vertical asymptote of the logarithmic function $y = \log_a(bx + c)$ occurs at $x = -\frac{c}{b}$.
Steps to Solve
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Identify the logarithmic function Begin by writing down the logarithmic function you are analyzing. For example, if we have the function $y = \log_a(bx + c)$, identify the values of $a$, $b$, and $c$.
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Set the argument of the logarithm to zero For vertical asymptotes, find where the function is undefined by setting the argument of the logarithm (the expression inside) to zero: $$ bx + c = 0 $$
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Solve for $x$ Rearranging the equation from the previous step gives: $$ bx = -c $$ Then divide both sides by $b$ (assuming $b \neq 0$): $$ x = -\frac{c}{b} $$
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Determine the vertical asymptote The value found in the previous step signifies the location of the vertical asymptote. For the function $y = \log_a(bx + c)$, the vertical asymptote occurs at: $$ x = -\frac{c}{b} $$
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State any horizontal asymptotes (if applicable) For logarithmic functions, horizontal asymptotes are generally not present. However, you can mention that as $x \to \infty$, $y \to \infty$, indicating no horizontal asymptote.
The vertical asymptote of the logarithmic function $y = \log_a(bx + c)$ occurs at $x = -\frac{c}{b}$.
More Information
Vertical asymptotes indicate where the function goes to infinity. For logarithmic functions, this typically represents points where the function cannot deliver a real output because the logarithm of zero or a negative number is undefined.
Tips
- Forgetting to set the argument of the logarithm equal to zero.
- Not checking that the base $b$ is not equal to zero when solving for $x$.
- Confusing vertical asymptotes with horizontal asymptotes.
