The document introduces exponential functions and the mathematical constant e. It discusses how e can be used to calculate interest payments split into multiple parts throughout a year. The value of e is approximately 2.71828. Graphs of the exponential function f(x)=ex and its inverse, the natural logarithm function f(x)=ln(x) are presented. Transformations of these functions are explored, including changing coefficients, horizontal and vertical shifts, and reflections. Solving equations involving exponentials and logarithms is also covered.

2. Introduction
• We are going to look at exponential
functions
• We will learn about a new ‘special’
number in Mathematics
• We will see how this number can be
used in practical problems…

3. The Exponential and Log Functions
 Imagine you have £100 in a bank account
 Imagine your interest rate for the year is 100%
 You will receive 100% interest in one lump at the end of the year, so you will now
have £200 in the bank
However, you are offered a possible alternative way of being paid
 Your bank manager says, ‘If you like, you can have your 100% interest split into
two 50% payments, one made halfway through the year, and one made at the end’
 How much money will you have at the end of the year, doing it this way (and
what would be the quickest calculation to work that out?)
 £100 x 1.52
 = £225
Investigate further. What would happen if you split the interest into 4, or 10, or
100 smaller bits etc…

4. The Exponential and Log Functions
£100e£100 x (1 + 1/n)n100/nn£100
£271.81£100 x 1.0001100000.01%10,000£100
£271.69£100 x 1.00110000.1%1,000£100
£270.48£100 x 1.011001%100£100
£269.16£100 x 1.02502%50£100
£265.33£100 x 1.05205%20£100
£259.37£100 x 1.11010%10£100
£256.58£100 x 1.125812.5%8£100
£244.14£100 x 1.25425%4£100
£225£100 x 1.5250%2£100
£200£100 x 2100%1£100
Total (2dp)Sum
Interest Each
Payment
Payments
Start
Amount
1
1
n
e
n
 
  
 
The larger the value of n, the better the accuracy of e…(2.718281828459…)

5. The Exponential and Log Functions
The mathematical constant e was invented by a Scottish scientist John
Napier and was first used by a Swiss Mathematician Leonhard Euler.
He introduced the number as a base of logarithms and started to use the
letter e when writing an unpublished paper on explosive forces in Cannons.
e, (Euler’s constant) is an irrational number whose value is e = 2.718281…
When mathematicians talk about exponential functions they are referring
to function ex, where e is the constant.

6. Graph of ex
The following diagram shows graph of y = ex
It is also known as an exponential graph.

7. You need to be able to sketch transformations of the graph y = ex
So lets recap our transformations
TASK 1: Match the equations to the graphs.
TASK 2: Fill in the table.
Describe what transformation is taking place.
Is it affecting the x or y?
Change the coordinates after the transformation.
The Exponential and Log Functions

8. You need to be able to sketch
transformations of the graph
y = ex
y = ex
y = 2ex
y = ex
(0,1)
f(x)
2f(x)
y = 2ex
(0,2)
The Exponential and Log Functions
(For the same set of
inputs (x), the
outputs (y) double)

9. You need to be able to sketch
transformations of the graph
y = ex
y = ex
y = ex + 2
y = ex
(0,1)
f(x)
f(x) + 2
y = ex + 2
(0,3)
The Exponential and Log Functions
(For the same set of
inputs (x), the
outputs (y) increase
by 2)

10. You need to be able to sketch
transformations of the graph
y = ex
y = ex
y = -ex
y = ex
(0,1)
f(x)
-f(x)
y = -ex
(0,-1)
The Exponential and Log Functions
(For the same set of
inputs (x), the
outputs (y) ‘swap
signs’

11. You need to be able to sketch
transformations of the graph
y = ex
y = ex
y = e2x
y = ex
(0,1)
f(x)
f(2x)
y = e2x
The Exponential and Log Functions
(The same set of
outputs (y) for half
the inputs (x))

12. You need to be able to sketch
transformations of the graph
y = ex
y = ex
y = ex + 1
y = ex
(0,1)
f(x)
f(x + 1)
y = ex + 1
The Exponential and Log Functions
(The same set of
outputs (y) for inputs
(x) one less than
before…)
(0,e)
We can work out the y-intercept by
substituting in x = 0
 This gives us e1 = e

13. You need to be able to sketch
transformations of the graph
y = ex
y = ex
y = e-x
y = ex
(0,1)
f(x)
f(-x)
y = e-x
The Exponential and Log Functions
(The same set of
outputs (y) for inputs
with the opposite
sign…
(0,1)

14. You need to be able to sketch
transformations of the graph
y = ex
Sketch the graph of:
y = 10e-x
y = ex
The graph of e-x,
but with y values 10
times bigger…
y = e-x
The Exponential and Log Functions
y = 10e-x
(0, 1)
(0, 10)

15. You need to be able to sketch
transformations of the graph
y = ex
Sketch the graph of:
y = 3 + 4e0.5x
y = ex
The graph of e0.5x,
but with y values 4
times bigger with 3
added on at the end…
(0, 1)
(0, 7)
y = e0.5x
y = 4e0.5x
y = 3 + 4e0.5x
(0, 4)
The Exponential and Log Functions

16. The Logarithmic Function
e is often used as a base for a logarithm.
This logarithm is called the natural logarithm
where logex is written as: ln x

17. Graph of ln x
The following diagram shows graph of y = ln x.
Note that the graph of
ln x does not exist in the
negative x-axis.
So, ln x does not exist
for negative value of x.
Try it on your
calculator now.

18. The Exponential and Log Functions
You need to be able to plot and
understand graphs of the
function which is inverse to ex
The inverse of ex is logex
(usually written as lnx)
y = ex
y = lnx
y = x
(0,1)
(1,0)

19. The Exponential and Log Functions
You need to be able to plot and
understand graphs of the
function which is inverse to ex
y = lnx
(1,0)y = lnx f(x)
y = 2lnx 2f(x)
y = 2lnx
All output (y) values
doubled for the same
input (x) values…

20. The Exponential and Log Functions
You need to be able to plot and
understand graphs of the
function which is inverse to ex
y = lnx
(1,0)y = lnx f(x)
y = lnx + 2 f(x) + 2
y = lnx + 2
(0.14,0)
ln 2y x 
0 ln 2x 
2 ln x 
2
e x

0.13533... x
Let y = 0
Subtract 2
Inverse ln
Work out x!
All output (y) values
increased by 2 for the
same input (x) values…

21. The Exponential and Log Functions
You need to be able to plot and
understand graphs of the
function which is inverse to ex
y = lnx
(1,0)
y = lnx f(x)
y = -lnx -f(x)
y = -lnx
All output (y) values
‘swap sign’ for the same
input (x) values…

22. The Exponential and Log Functions
You need to be able to plot and
understand graphs of the
function which is inverse to ex
y = lnx
(1,0)
y = ln(x) f(x)
y = ln(2x) f(2x)
y = ln(2x)
All output (y) values the
same, but for half the
input (x) values…
(0.5,0)

23. The Exponential and Log Functions
You need to be able to plot and
understand graphs of the
function which is inverse to ex
y = lnx
(1,0)
y = ln(x) f(x)
y = ln(x + 2) f(x + 2)
y = ln(x + 2)
All output (y) values the
same, but for input (x)
values 2 less than
before
(-1,0)
ln( 2)y x 
ln(2)y 
0.69314...y 
Let x = 0
Work it out (or
leave as ln2)
(0, ln2)

24. The Exponential and Log Functions
You need to be able to plot and
understand graphs of the
function which is inverse to ex
y = lnx
(1,0)
y = ln(x) f(x)
y = ln(-x) f(-x)
y = ln(-x)
All output (y) values the
same, but for input (x)
values with the opposite
sign to before
(-1,0)

25. The Exponential and Log Functions
You need to be able to plot and
understand graphs of the
function which is inverse to ex
y = lnx
(1,0)
Sketch the graph of:
y = 3 + ln(2x)
y = ln(2x)
(0.025,0)
y = 3 + ln(2x)
The graph of ln(2x),
moved up 3 spaces…
3 ln(2 )y x 
0 3 ln(2 )x 
3 ln(2 )x 
3
2e x

3
2
e
x


Let y = 0
Subtract 3
Reverse ln
Divide by 2

26. The Exponential and Log Functions
You need to be able to plot and
understand graphs of the
function which is inverse to ex
y = lnx
(1,0)
Sketch the graph of:
y = ln(3 - x)
The graph of ln(x),
moved left 3
spaces, then
reflected in the y
axis. You must do
the reflection last!
y = ln(3 + x)
(-2,0)
(2,0)
y = ln(3 - x)
ln(3 )y x 
ln(3)y 
Let x = 0
(0,ln3)

27. Describe the transformations of the lnx and ex functions.
Draw the old and new curves
(on the same graph) for each question.
The Exponential and Log Functions

29. What’s the connection?
You already know that every logarithmic function
has an inverse involving an exponential function.
e.g. log2x = 5  x = 25
Hence, the natural log, ln x has an inverse of
the exponential function ex.

30. The Exponential and Log Functions
You need to be able to
solve equations involving
natural logarithms and e
 This is largely done in the same
way as in C2 logarithms, but using
‘ln’ instead of ‘log’
Example Question 1
3x
e 
ln( ) ln(3)x
e 
ln( ) ln(3)x e 
ln(3)x 
1.099x 
Take natural logs of
both sides
Use the ‘power’ law
ln(e) = 1
Work out the answer or
leave as a logarithm
You do not necessarily
need to write these
steps…

31. The Exponential and Log Functions
You need to be able to
solve equations involving
natural logarithms and e
 This is largely done in the same
way as in C2 logarithms, but using
‘ln’ instead of ‘log’
2
7x
e 

Take natural logs
Use the power law
2
ln( ) ln(7)x
e 

2 ln(7)x  
ln(7) 2x  
0.054x  
Subtract 2
Work out the answer or
leave as a logarithm
TRY THIS ONE

32. The Exponential and Log Functions
You need to be able to
solve equations involving
natural logarithms and e
 This is largely done in the same
way as in C2 logarithms, but using
‘ln’ instead of ‘log’
ln(3 2) 3x  
‘Reverse ln’
Add 2
3
3 2x e 
3
3 2x e 
3
2
3
e
x


Divide
by 3
Example Question 2

33. The Exponential and Log Functions
You need to be able to
solve polynomials involving
natural logarithms and e 2
(ln ) 3ln 2 0x x  
Example Question 3
Solve
 These equations always
look scary, however they
are only disguised cubic or
quadratic equations (which
you can solve!)

34. The Exponential and Log Functions
You need to be able to
solve polynomials involving
natural logarithms and e 3 2
2 2 0y y y
e e e   
Example Question 4
Solve
 These equations always
look scary, however they
are only disguised cubic or
quadratic equations (which
you can solve!)

35. The Exponential and Log Functions
You need to be able to
solve polynomials involving
natural logarithms and e MIXED EXERCISE Page 75
Question 1 (any 2 parts)
Question 2 (any 3 parts)
Question 3 (any 3 parts)
 These equations always
look scary, however they
are only disguised cubic or
quadratic equations (which
you can solve!)

37. Differentiation of ex
ex is the only function which is
unchanged when differentiated.
i.e.
x
y e
xdy
e
dx


38. Differentiation of ex
x
y e
xdy
e
dx

kx
y e
kxdy
ke
dx


39. The Exponential and Log Functions
You need to be able to
differentiate exponential
functions
Examples
a. y = e-3x
b. y = x5 – e4x
c. y = e2x + 3
d. y =
5
2
5x
x
e
e

Exercise 4A Page 64
Question 3, 5, 6, 7

40. Differentiation of ln x
Using the exponential function ex it can be
proved that the differentiation of ln x is 1
x
1dy
dx x
lny x

41. Proof of Differentiation of ln x
Consider the function y = ln x y = loge x
In exponential form: x = ey
Consider, as a differentiation of ln x with respect to x.
As, x = ey
This means, Proved.
dy
dx
dy
dx
=1¸
dx
dy
dx
dy
= ey
dx
x
dy

1
dy
x
dx
 

42. Examples
Find for each of the following:
a. y = x2 – ln(3x)
b. y = 5x3 – 6lnx + 1
dy
dx
Exercise 4E Page 72
Question 1, 7 and 9

43. Integration of ex
x
y e c xdy
e
dx

1 kx
y e c
k
 kxdy
e
dx


44. lny x c 
1dy
dx x

Integration of lnx

45. Examples
Integrate the following:
a) y = e3x
b) y = e½x
c)
3
2x
y
x


Exercise 4B Page 66
Question 4, 5, 6
Exercise 4D Page 70
Question 6, 7
Exercise 4E Page 73
Question 2, 6, 8

46. Mixed Exercise
EXAM QUESTIONS
Page 76
Question 6, 7, 8, 10