So recall we introduced the natural logarithm. Integrals interpret the natural logarithm as , and we will explore that in this post.
1. The natural logarithm satisfies
By the fundamental theorem of calculus, we can consider a different definition
Observe that, by (1), the first derivative is positive for . Also by differentiating it again, we get
Thus is concave down but increasing. Thus its sketch should look like:
So What?!? Well, we have an effective method to calculate the natural logarithm of any number! As a matter of fact, we can show a doodle what the logarithm corresponds to:
The curve is drawn is blue, and the area between and is shaded in light blue. That light blue region’s area corresponds to the natural logarithm.
WARNING: The blue curve doodled is the derivative of the natural logarithm. This is a subtle point which some students miss, and believe the blue curve is the natural logarithm. This tempting mistake must be avoided!
2. What about ? Recall that is the number satisfying
The integral form of this statement is
We can doodle this situation as
We want the shaded green region to have an area equal to 1.