Question
What is the range of the logarithm function defined as \( g(x) = \log_{10}(x) \)?
Ask by Chandler Park. in Zambia
Feb 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The range of \( g(x) = \log_{10}(x) \) is all real numbers.
Solution
To determine the range of the logarithm function defined as \( g(x) = \log_{10}(x) \), we can analyze the behavior of the function.
1. **Domain of the function**: The logarithm function \( g(x) = \log_{10}(x) \) is defined for \( x > 0 \). This means that the input \( x \) must be a positive real number.
2. **Behavior of the function**:
- As \( x \) approaches \( 0 \) from the right (i.e., \( x \to 0^+ \)), \( g(x) \) approaches \( -\infty \).
- As \( x \) increases towards \( +\infty \), \( g(x) \) also increases without bound, approaching \( +\infty \).
3. **Conclusion**: Since \( g(x) \) can take any real number value as \( x \) varies over its domain, the range of the function \( g(x) = \log_{10}(x) \) is all real numbers.
Thus, the range of \( g(x) = \log_{10}(x) \) is:
\[
\text{Range} = (-\infty, +\infty)
\]
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Simplify this solution Mind Expander
The range of the logarithm function \( g(x) = \log_{10}(x) \) is all real numbers, which can be expressed as \( (-\infty, \infty) \). This means that as \( x \) approaches 0 from the right, \( g(x) \) tends toward negative infinity, and as \( x \) increases to infinity, \( g(x) \) also increases without bound. In practical applications, understanding the range of the logarithm function is crucial, especially in fields such as science and finance, where it can be used to analyze exponential growth or decay. For example, in measuring sound intensity (decibels) or the pH of solutions, the logarithm helps to convert multiplicative relationships into more manageable additive ones!
