Question

Which one of the following observations is correct for the features of the logarithm function to any base $b > 1$?

• A. The domain of the logarithm function is $\mathbb{R}$, the set of real numbers.
• B. The range of the logarithm function is $\mathbb{R}^+$, the set of all positive real numbers.
• C. The point $(1, 0)$ is always on the graph of the logarithm function.
• D. The graph of the logarithm function is decreasing as we move from left to right.

Answer 1

The Correct Option is C

1. Understand the problem:

We need to identify the correct statement about the features of the logarithm function for any base \( b > 1 \).

2. Analyze each option:

(A) The domain of the logarithm function is \( \mathbb{R} \). Incorrect. The domain is \( (0, \infty) \).

(B) The range of the logarithm function is \( \mathbb{R}^+ \). Incorrect. The range is \( \mathbb{R} \).

(C) The point \( (1, 0) \) is always on the graph of the logarithm function. Correct. For any base \( b > 1 \), \( \log_b 1 = 0 \).

(D) The graph of the logarithm function is decreasing as we move from left to right. Incorrect. The graph is increasing for \( b > 1 \).

Correct Answer: (C) The point (1, 0) is always on the graph of the logarithm function.

Answer 2

• (A) The domain of the logarithm function is \( \mathbb{R} \), the set of real numbers. This is incorrect. Logarithms are only defined for positive real numbers.
• (B) The range of the logarithm function is \( \mathbb{R}^+ \), the set of all positive real numbers. This is incorrect. The range of a logarithm function is all real numbers (\( \mathbb{R} \)).
• (C) The point \( (1, 0) \) is always on the graph of the logarithm function. This is correct. Because \( b^0 = 1 \) for any base \( b > 0 \), \( \log_b(1) = 0 \). Therefore, the point \( (1, 0) \) will always lie on the graph.
• (D) The graph of the logarithm function is decreasing as we move from left to right. This is incorrect. When the base \( b > 1 \), the logarithm function is increasing.

Therefore, the answer is \( \boxed{C} \).