What are the domain and range of ln(x)?
Understand the Problem
The question is asking for the domain and range of the natural logarithm function, denoted as ln(x). To solve this, we need to analyze the function to determine the values of x for which ln(x) is defined (domain) and the possible output values of the function (range).
Answer
Domain: $(0, \infty)$; Range: $(-\infty, \infty)$
Answer for screen readers
The domain of the natural logarithm function is $(0, \infty)$ and the range is $(-\infty, \infty)$.
Steps to Solve
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Determine the Domain The domain of the natural logarithm function $f(x) = \ln(x)$ is the set of all real numbers $x$ for which the function is defined. Since $\ln(x)$ is only defined for positive values of $x$, the domain is: $$ x > 0 $$ Therefore, the domain in interval notation is: $$ (0, \infty) $$
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Determine the Range Next, we need to find the range of the function $f(x) = \ln(x)$. The natural logarithm can take on all real values, which means that as $x$ approaches 0 from the right, $\ln(x)$ approaches $-\infty$, and as $x$ increases to $\infty$, $\ln(x)$ approaches $\infty$. Thus, the range of the function is: $$ (-\infty, \infty) $$
The domain of the natural logarithm function is $(0, \infty)$ and the range is $(-\infty, \infty)$.
More Information
The natural logarithm is an important function in calculus and is often used in solving equations involving exponential growth, natural growth, and in many areas of science and engineering.
Tips
Common mistakes include:
- Assuming the natural logarithm is defined for $x \leq 0$. Remember that $\ln(x)$ is only defined for positive $x$.
- Forgetting that the range extends to both positive and negative infinity.
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