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The image presents a question about the domain and range of the function \( f(x) = \log_b x \) where \( b > 0 \) and \( b \neq 1 \).

The correct domain for \( f(x) = \log_b x \) is \( D = (0, \infty) \). This is because the logarithm function, \( \log_b x \), is defined only for positive values of \( x \). The values of \( x \) must be greater than 0 to satisfy the conditions for a valid logarithmic input. 

The choice in the image labeled as option B, "D = (0, \infty), because the range of its inverse function \( g(x) = b^x \) is \( (0, \infty) \)," is the correct answer. The reason provided is also accurate: the domain of a function is the range of its inverse, and vice versa. The inverse function of \( f(x) = \log_b x \) is \( g(x) = b^x \), which indeed has a range of \( (0, \infty) \) since \( b^x \) yields positive outputs for any real \( x \).

Do you need any further clarification on this concept, or is there another question on this topic you would like to explore? Here are some related questions you might consider:

1. What is the range of \( f(x) = \log_b x \)?
2. How does changing the base \( b \) of the logarithm affect the graph of \( f(x) = \log_b x \)?
3. What are the properties of the logarithm function that are important for solving equations?
4. How do you solve equations involving logarithms?
5. Can you explain how to convert between different logarithmic bases?

**Tip:** Remember, understanding the relationship between a function and its inverse can greatly simplify solving problems related to domain and range.

For b > 0 with b ≠ 1, what are the domain and range of f(x) = log_b x and why?

Learn how to determine the domain and range of the logarithmic function f(x) = log_b x, where b > 0 and b ≠ 1. This problem covers core concepts in logarithmic functions, inverse functions, and understanding domain and range relationships.

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