Exponentials and logarithms

Higher
Mathematics
EXPONENTIALS
LOGARITHMS &


Contents
Exponentials and Logarithms 1
1 Exponentials EF 1
2 Logarithms EF 3
3 Laws of Logarithms EF 3
4 Exponentials and Logarithms to the Base e EF 6
5 Exponential and Logarithmic Equations EF 7
6 Graphing with Logarithmic Axes EF 10
7 Graph Transformations EF 14

Exponentials
Logarithmsand
1 Exponentials
EF

We have already met exponential functions in the notes on
Functions and Graphs..

If a  1ytheny the
 axgraph
, a looks
1 like this:



1, a 
1

O

, This is sometimes called a growth
function.
x


If 0  a  1 then the graph looks like this:
y
y  ax , 0  a  1

1 1, a 

O


This is sometimes called a decay function.

x

Remember that the graph of an exponential function

f  x   ax

always
passes through 0, 1
and 1, a 

since:

f 0  a0  1, f 1  a1  a
.

Let u0 be the initial population.

u1  1·16u0


(116% as a decimal)

, u2  1·16u1 
1·161·16u0  
1·162u0 u3 
1·16u2 
1·161·162u0  
1·163u0

un  1·16n u0.
For the population to double after n years, we require un  2u0 .
We want to know the smallest n which gives 1·16n a value of 2 or
more,
since this will make un at least twice as big as u0 .
Try values of n until this is satisfied.
O 6 
n

a

c
a
l
c
u
l
a
t
o
r
:
1 ANS 



If n 
2, 1·162
 1·35
 2 If n
 3,
1·163 
1·56  2
If n 
4, 1·164
 1·81
2

, If n  5, 1·165  2·10  2
Therefore after 5 years the population will double.


Let u0

be the initial efficiency.

u1  0·95u0 (95% as a decimal)

u2  0·95u1  0·950·95u0   0·952u0 u3  0·95u2 
0·950·952u0   0·953u0

un  0·95n u0.
When the efficiency drops below 0·75u0




(75% of the initial value) the
machine must be serviced. So the machine needs serviced after n
years if
0·95n  0·75.
Try values of n until this is satisfied:

If n  2, 0·952  0·903 If n  3, 0·953  0·857 If n  4, 0·954 
0·815 If n  5, 0·955  0·774 If n  6, 0·956  0·735

 0·75
 0·75
 0·75
 0·75
 0·75
Therefore after 6 years, the machine will have to be serviced.
2 Logarithms
EF

Having previously defined what a logarithm is (see the
notes on Functions and Graphs) we now look in more
detail at the properties of these functions.