and natural logs

Aims: to know the value of e and how this relates to natural logs.

To be able to sketch graphs of ex and lnxTo start to solve equations involving e and ln

Write the equivalent exponential statement and state the value.

8log2

91log3

1000log

41log2 2

49log7log 77

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 ex

• 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!

Finding e!• Enter graph mode on

your calculator.• Draw y=2x

• (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 3x

• Consider your results.

–6 –4 –2 2 4 6

5

10

15

20

x

y

–6 –4 –2 2 4 6

5

10

15

20

x

y

xy 2

xy 3–6 –4 –2 2 4 6

5

10

15

20

x

y

–6 –4 –2 2 4 6

5

10

15

20

x

y

xy 2

xy 3

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.

The value of e

2.71828182845904523536028747135266249775724709369995.............................................................................................

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.

Remembering e• To• Express• e• Remember• To• Memorise• A• Sentence• To• Simplify• You can then think of a right angled

isosceles triangle angles 450, 900, 450 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.

Something else to consider..• If y = ex then x =

• loge is the natural log written as ln

• ex has a special property…• …the value of ex is the same as the gradient of the curve at that point.

• y = ex then dy/dx = ex

Considering the GraphsTasks• Sketch ex

• 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.

Laws of Logs Reminder All the laws of logs apply to ln! lna = c ec = a (inverse operations) lna + lnb = lna – lnb = lnac= lne = ln1= lnec = elna =

3log3ln!Re

e

shorthandthemember

A Few Quick Calculations

1) eln4 2) lne7 3) 5lne-0.1 4) e½ln9

ln(5+2x) -10 = 0

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……