Science:Math Exam Resources/Courses/MATH110/April 2012/Question 02 (b)/Solution 1

If you draw a picture of $\ln(x)$ and the linear approximation $y=x-1$ (see below), you can see that the linear approximation is above the graph of $\ln(x)$. This means that the value of the linear approximation will be greater than the value of the actual function.

Another way of showing this is to prove that $\ln(x)$ is concave down. A concave down function has all of its tangent lines above the graph of the function and so the same conclusion follows as above. We know $\ln(x)$ is concave down because its second derivative is $-1/x^{2}$ which is always negative.