What is the range of the logarithm function defined as \( g(x) = \log_{10}(x) \)?

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: \[ \text{Range} = (-\infty, +\infty) \]