1. Aims: to know the value of e and how this relates to natural logs. To be able to sketch graphs of e x and lnx To start to solve equations involving e and ln

2. Write the equivalent exponential statement and state the value.

3. What is e? e is an irrational number like π e is the value where the gradient of the curve has the same value as e x This proves to be really useful in lots of area of mathematics. It also allows us to differentiate exponential functions something we will look at in a few lessons time. We are going to investigate the value of e now!

4. Finding e! Enter graph mode on your calculator. Draw y=2 x (SHIFT) (MENU) takes you into the SET UP options. Change derivative to ON. If you press trace you can move the cursor along the curve. Note down the value of dy/dx at key points. Repeat for 3 x Consider your results.

5. Finding e! - Investigation Keep changing the value of the base number on your calculator. You are trying to get the value of dy/dx as close to the y value as possible. If you can get the two the same or close you will have found an approximation for e. No cheating! Try to record your results in a systematic way.

6. The value of e 2.71828 1828459045235 36028747135266249775724709 369995.............................................................................................

7. Other ways to find e The value of e is also equal to 1 + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + 1/7! +... (etc) For example, the value of (1 + 1/n) n approaches e as n tends towards infinity.

8. Remembering e To Express e Remember To Memorise A Sentence To Simplify You can then think of a right angled isosceles triangle angles 45 0, 90 0, 45 0 to get 6 more numbers! Count the number of letters to get e to 9 d.p. Remember the triangle below for 6 more! You now know e to 15d.p.

9. Something else to consider.. If y = e x then x = log e is the natural log written as ln e x has a special property… …the value of e x is the same as the gradient of the curve at that point. y = e x then dy / dx = e x

10. Considering the Graphs Tasks Sketch e x Sketch lnx on the same axes Can you spot anything interesting? Using your knowledge of transformations from Core 2 and FP1 complete the matching cards.

11. Laws of Logs Reminder

12. A Few Quick Calculations

13. Benjamin Peirce (1809-1880, American mathematician, professor at Harvard) gave a lecture proving "Euler's equation", and concluded: "Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth.” And Finally……