The natural logarithm function ln(x) is concave on its domain. True or false?

The natural logarithm function ln(x) is concave on its domain. True or false?

• A. True
• B. False
• C. Both
• D. Neither

Answer: (A)

The key idea here is concavity and how it shows up through the second derivative. For f(x) = ln x, the first derivative is f'(x) = 1/x and the second derivative is f''(x) = -1/x^2. Since x must be positive in the domain of the natural logarithm, f''(x) is negative for every point in that domain. A negative second derivative means the graph curves downward everywhere on its domain, so the function is concave there. The natural logarithm is increasing, but its slope 1/x decreases as x grows, which aligns with concavity. Therefore, the statement that ln(x) is concave on its domain is true.

The key idea here is concavity and how it shows up through the second derivative. For f(x) = ln x, the first derivative is f'(x) = 1/x and the second derivative is f''(x) = -1/x^2. Since x must be positive in the domain of the natural logarithm, f''(x) is negative for every point in that domain. A negative second derivative means the graph curves downward everywhere on its domain, so the function is concave there. The natural logarithm is increasing, but its slope 1/x decreases as x grows, which aligns with concavity. Therefore, the statement that ln(x) is concave on its domain is true.