3 The Exponential and
Logarithmic Function
Contents
Section ..................................................................................................................................... Page
3 The Exponential and Logarithmic Function ............................................................................. 1
3.1 Growth and Decay .................................................................................................................. 1
Note 1 – Growth and Decay...................................................................................................... 1
Example 2 – Growth ................................................................................................................. 2
Example 3 – Decay ................................................................................................................... 2
Further Examples ..................................................................................................................... 2
3.2 A Special Exponential Function exp(x) ................................................................................... 3
Note 1 – The Exponential Function ex ...................................................................................... 3
Example 2 – Calculating Continuous Growth............................................................................ 4
Further Examples ..................................................................................................................... 4
3.3 Exponential Functions and Related Logarithmic Functions .................................................... 5
Definition 1 – The Relationship between Exponential Functions and Logarithmic Functions .... 5
Note 2 – Some Important Results ............................................................................................. 5
Example 3 – Converting between exponential form and logarithmic form ................................ 5
Further Examples ..................................................................................................................... 6
3.4 Rules of Logarithms................................................................................................................ 8
Definition 1 – Rules of Logarithms ............................................................................................ 8
Example 2 – Using the Rules of Logarithms ............................................................................. 8
Example 2 – Solving Simple of Logarithmic Equations ............................................................. 9
Further Examples ..................................................................................................................... 9
3.5 Solving Exponential Equations ............................................................................................. 10
Example 1 – Solving Exponential Equations........................................................................... 10
Further Examples ................................................................................................................... 10
Page i
3 The Exponential and Logarithmic Function
3.1 Growth and Decay
3.1 Growth and Decay
Note 1 – Growth and Decay
The growth of money left in a bank account, the population of a country and the reproduction of
cells are example of exponential growth and can all be modelled by an exponential function.
y
Exponential Growth Function
x
The decay of radioactive elements and the decrease in atmospheric pressure with height above
sea level are examples of exponential decay and can be modelled by an exponential function.
y
Exponential Decay Function
x
Page 1
3 The Exponential and Logarithmic Function
3.1 Growth and Decay
Example 2 – Growth
Scotland’s population is currently 5.2 million and is growing at a rate of 0.54% per year.
(a)
Construct a growth function for the population in millions n years later.
(b)
Estimate the population of Scotland in 30 years time?
Example 3 – Decay
The half-life of strontium 90 is 29 years.
(a)
Write down the decay function for 5kg of strontium 90 after t years.
(b)
Calculate the mass of strontium 90 after 45 years to 2 decimal places.
Further Examples
Maths In Action
Page 236
Exercise 1/2
Page 2
3 The Exponential and Logarithmic Function
3.2 A Special Exponential Function exp(x)
3.2 A Special Exponential Function exp(x)
Note 1 – The Exponential Function ex
The exponential function is defined as e x , sometimes written as exp( x ) , where e is an irrational
number like π . The number e  2.71828182845905 to 14 decimal places.
This number arises naturally in many areas of mathematics and science.
Consider compounding interest (interest = 100%):
1
Yearly
 1
1 1  2


Semi-annually
1

 1  2   2.25


Quarterly
1

 1  4   2.44140625


Monthly
1

 1  12   2.61303529022...


Weekly
1 

 1  52 


Daily
1 

 1  365 


Hourly
1 

 1  8760 


Every Minute
1


 1  525600 


Every Second
1


 1  31536000 


2
4
12
52
 2.69259695444...
365
 2.71456748202...
8760
 2.71812669063...
525600
 2.7182792154...
31536000
 2.71828247254...
Notice that the more the interest in compounded, the closer to e the number becomes. If interest is
calculated continuously then the formula for calculating the amount of money A, after time t is
A  A0ert where r is the interest rate and A0 is the initial amount of money.
Page 3
3 The Exponential and Logarithmic Function
3.3 Exponential Functions and Related Logarithmic Functions
Example 2 – Calculating Continuous Growth
Due to inflation, prices increase annually. The price, P, of items after t years is given by
P(t )  P0e0.045t , where P0 is the original price (the inflation rate is 4.5%).
If a bottle of irn bru cost 84p then how much will it cost in:
(a)
2 years
(b)
10 years?
Further Examples
Maths In Action
Page 239
Exercise 3
Q6-9
Page 4
3 The Exponential and Logarithmic Function
3.3 Exponential Functions and Related Logarithmic Functions
3.3 Exponential Functions and Related Logarithmic Functions
Definition 1 – The Relationship between Exponential Functions and Logarithmic Functions
An exponential function is related to a logarithmic function by y  a x  loga y  x . Note the base
of the logarithm function a is the same as the number raised to the power of x for the exponential
function.
Note 2 – Some Important Results
Since a0  1 then loga 1  0 . Similarly, a1  a then loga a  1.



a0  1
a1  a
loga 1  0
a number raised to the power of 0 equals 1
a number raised to the power of 1 is that number
log 1 to any base is 0

loga a  1
the log of a number to that base is 1
Example 3 – Converting between exponential form and logarithmic form
1.
Write in logarithmic form:
(a)
2.
 24
(b)
1
16
(b)
log27 9 
Write in exponential form:
(a)
3.
8  23
log3 81  4
Solve logx 9  2 .
Page 5
2
3
3 The Exponential and Logarithmic Function
3.3 Exponential Functions and Related Logarithmic Functions
4.
Find the value of log4 16 .
Further Examples
Maths In Action
Page 240
Exercise 4
Page 6
3 The Exponential and Logarithmic Function
3.3 Exponential Functions and Related Logarithmic Functions
Page 7
3 The Exponential and Logarithmic Function
3.4 Rules of Logarithms
3.4 Rules of Logarithms
Definition 1 – Rules of Logarithms
1.
loga xy  loga x  loga y
2.
x
loga    loga x  loga y
y
3.
loga x p  p loga x
Example 2 – Using the Rules of Logarithms
Simplify the following expressions:
(a)
log3  log7
(b)
log8  log4
(c)
log8  log3  log6
(d)
log27  3 log3
(e)
log2 4  log2 4
(f)
3 log3 2  21 log3 4
(g)
log5 25
(h)
log2 32
Page 8
3 The Exponential and Logarithmic Function
3.4 Rules of Logarithms
Example 3 – Solving Simple of Logarithmic Equations
Solve
(a)
loga x  loga 3  loga 15
(b)
Further Examples
Maths In Action
Page 242
Exercise 5
Page 9
loga x  2 loga 4  loga 2
3 The Exponential and Logarithmic Function
3.5 Solving Exponential Equations
3.5 Solving Exponential Equations
Example 1 – Solving Exponential Equations
(a)
Solve 4x  100 to 2 decimal places.
(b)
The value of a house V (t ) after t years is given by V (t )  120000(1.04)t .
After how many years will the value exceed £150,000?
Further Examples
Maths In Action
Page 244
Exercise 6
Page 10