Exam (elaborations)

Exponentials and logarithms

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Course
Mathematics

Institution
Mathematics

Exponentials and logarithms are inverse operations that use the same information but differ in what they find12. An exponential is used to find the value of the base raised to an exponent, whilst a logarithm is used to find the exponent (power)1. The logarithm tries to lead you to the exponent need...

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exponentials and logarithms

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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.

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