Is the natural logarithm function ln(x) increasing for x > 0?

Is the natural logarithm function ln(x) increasing for x > 0?

• A. True
• B. False
• C. It increases but at a decreasing rate
• D. It decreases

Answer: (A)

For x > 0, ln(x) increases as x grows. This happens because its slope is positive there: the derivative of ln(x) is 1/x, which is greater than zero for all positive x. So moving to the right along the x-axis yields higher ln values, meaning the function rises. The positive slope also explains why the rate of growth slows as x increases—the derivative 1/x gets smaller with larger x, so the curve is concave down. That’s why you can also describe its growth as increasing but at a decreasing rate, but the essential point for the question is that it does increase. It does not decrease or stay constant on x > 0.

For x > 0, ln(x) increases as x grows. This happens because its slope is positive there: the derivative of ln(x) is 1/x, which is greater than zero for all positive x. So moving to the right along the x-axis yields higher ln values, meaning the function rises. The positive slope also explains why the rate of growth slows as x increases—the derivative 1/x gets smaller with larger x, so the curve is concave down. That’s why you can also describe its growth as increasing but at a decreasing rate, but the essential point for the question is that it does increase. It does not decrease or stay constant on x > 0.