1. Base e and Natural Logarithms

2. History The number e is a famous irrational number, and is one of the most important numbers in mathematics. The first few digits are 2.7182818284590452353602874713527... It is often called Euler's number after Leonhard Euler. e is the base of the natural logarithms (invented by John Napier).

3. Calculating The value of (1 + 1/n) n approaches e as n gets bigger and bigger: n (1 + 1/n) n 12.00000 22.25000 52.48832 102.59374 1002.70481 1,0002.71692 10,0002.71815 100,0002.71827

4. Vocabulary natural base: the number e, which is found using the base rate of growth shared by all continually growing processes Used heavily in science to model quantities that grow & decay continuously natural base exponential function: an exponential function with base e

5. Vocabulary natural logarithm: a logarithm with base e The natural log gives you the time needed to reach a certain level of growth. natural logarithmic function: the inverse of the natural base exponential function

6. Ex 1 Use a calculator to estimate to four decimal places. Ex 2 Ex 3 Ex 4

7. Ex 5 Exponential  logarithmic Write an equivalent equation in the other form. Ex 6 Writing Equivalent Expressions Ex 8 Ex 7

8. Inverse Properties

9. Ex 9 Evaluate Ex 11 Writing Equivalent Expressions Evaluate Ex 10 Evaluate Ex 12

10. Solving Equations Ex 13 Solve the following equations. Ex 14

11. Solving Equations Ex 15 Solve the following equations. Ex 16

12. Graphing properties ( and lnx are inverse functions reflected in y=x) These can be transformed (stretched, translated and reflected) in the same way as other functions. But, be careful of the hidden asymptotes!

13. Examination-style question The function f is defined by a)Describe the sequence of geometrical transformations by which the graph of y = 3 e x + 1 – 4 can be obtained from that of y = e x. b)The graph of y = f ( x ) crosses the y -axis at point A and the x -axis at point B. Write down the coordinates of A and B, working to 2 decimal places. c)Write an expression for f –1 ( x ) and state its domain and range. d)Sketch the graphs of y = f ( x ) and y = f –1 ( x ) on the same set of axes and state their geometrical relationship.

14. Examination-style question d) y = f –1 ( x ) is a reflection of y = f ( x ) in the line y = x. y = x x y y = –4 y = f ( x ) y = f –1 ( x ) x = –4 (–0.71, 0) (0, –0.71) (0, 4.15) (4.15, 0)