How to find the range of a log function?
Understand the Problem
The question is asking how to determine the range of a logarithmic function, which involves understanding the properties of logarithms and the values that the function can output.
Answer
The range of the logarithmic function $f(x) = \log_b(x)$ is $(-\infty, \infty)$.
Answer for screen readers
The range of the logarithmic function $f(x) = \log_b(x)$ is $(-\infty, \infty)$.
Steps to Solve
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Identify the function First, we need to clearly state the logarithmic function we are dealing with, for example, $f(x) = \log_b(x)$, where $b$ is the base of the logarithm.
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Determine where the function is defined Logarithmic functions are defined only for positive values of their argument. Therefore, $x$ must be greater than 0: $$ x > 0 $$
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Analyze the output values The output (or range) of the logarithmic function can be thought of in terms of how the function behaves. As $x$ approaches $0$ from the right, $f(x)$ approaches $-\infty$, and as $x$ increases towards infinity, $f(x)$ approaches $\infty$.
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Conclusion about the range Since the function can output values from $-\infty$ to $\infty$, we summarize the range of the logarithmic function: $$ \text{Range } f(x) = (-\infty, \infty) $$
The range of the logarithmic function $f(x) = \log_b(x)$ is $(-\infty, \infty)$.
More Information
Logarithmic functions are unique because they can take any positive value of $x$ and will always yield a real number output. This property makes them very useful in many areas of mathematics, including solving equations and modeling growth.
Tips
- A common mistake is thinking that the logarithm can take zero or negative values; remember that $\log_b(x)$ is only defined for $x > 0$.
- Confusing the domain and range of the logarithmic function can lead to errors. Always remember that the domain affects the input, while the range describes possible outputs.
