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

The logarithm function $\log_b(x)$ satisfies the following:

1. Domain: $x > 0$, so it is not $\mathbb{R}$. 

. Range: $\mathbb{R}$, not $\mathbb{R}^+$. 

3. Point on Graph: $\log_b(1) = 0$, so the point $(1, 0)$ is always on the graph. 

4. Monotonicity: The graph is increasing for $b > 1$. 

Hence, the correct observation is that the point $(1, 0)$ is always on the graph of the logarithm function.