1. The Natural Log Function
5.1 Differentiation
5.2 Integration

2. A Brief History of e
John Napier, Scottish, Inventor of Logarithms, e implied in his work—gave table
of natural log values (although did not recognize
e as base)
1661 Christian Huygens, Dutch, studies relationship between the area under a rectangular hyperbola and logarithms, but does not see connection to e (later he does evaluate log e to 17 decimal places, but does see that this is a log)
1668 Nicolaus Mercator, German, names the natural logarithm, but does not discuss its base

3. A Brief History of e
1683 Jacques Bernoulli, Swiss, discovers e through study of compound interest, does not call it e or recognize its connections to logs
1697 Johann Bernoulli (Jacques brother) begins study of the calculus of exponential functions and is perhaps the first to recognize logs as functions
1720s Leonard Euler, Swiss, first studied e, proved it irrational, and named it (the fact that it is the first letter of his surname is coincidental).

4. A Brief History of Logs
Napier: studies the motion of someone covering a distance d whose speed at each instant is equal to the remaining distance to be covered. He divided the time into short intervals of length  , and assumed that the speed was constant within each short interval. He tabulated the corresponding values of distance and time obtained in this way.
He coined a name for their relationship out of the Greek words logos (ratio) and arithmos (number). He used a Latinized version of his word: logarithm.
In modern terms, we can say that the base of the logarithm in Napier's table was
The actual concept of a base was not developed until later.

5. Definition of Natural Log Function
1647--Gregorius Saint-Vincent,
Flemish Jesuit, first noticed that
the area under the curve from 1 to e is 1, but does not define or recognize the importance of e.

6. Definition of Natural Log Function

7. Definition of e

8. Review: Properties of Logs (i.e. Making life easier!)

9. Practice: Expand each expression

10. Derivatives of the Natural Log Function

11. Practice: Find f ’ (Don’t forget your chain rule!!)

16. Log Rule for Integration

17. Practice: Don’t forget to use u-substitution when needed.

23. Integrals of the Trig Functions
** Prove these!