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Higher Maths
Unit 3 Chapter 3
Logarithms
Experiment & Theory
In experimental work
Introduction
data can often be modelled by equations
of the form:
we often want to create a mathematical model
ny ax xy ab
Polynomial function Exponential function
ny ax
Polynomial function
xy ab
Exponential function
Often it is difficult to know which model to choose
A useful way is to take logarithms
(i) For ny ax
log log ny ax
log log log ny a x
log log logy a n x
This is like logY a nX
or rearranging logY nX a Y mX c
A useful way is to take logarithms
(ii) For xy ab
log log xy ab
log log log xy a b
log log logy a x b
This is like log logY a x b
or rearranging log logY b x a Y mx c
ny ax
logY nX a
log log logy a n x
Plotting log y against log x gives us a straight lineIn the case ofa simple polynomial
xy ablog log logy a x b
log logY b x a
In the case ofAn exponential Plotting log y against x gives us a straight line
By drawing the straight line graph
The constants for
the gradient m
the y-intercept c can be found
log log logy a n x ny ax
xy ab log log logy a x b
c m
c m
1.0 1.62.0
2.3
x
y
2.1
2.2
1.1 1.2 1.3 1.4 1.5
Example The table shows the result of an experiment
x 1.1 1.2 1.3 1.4 1.5 1.6
y 2.06 2.11 2.16 2.21 2.26 2.30
How are x and y related ?
A quick sketch
x
x
x
x
x
x
suggests
ny ax
Now take logarithms of x and y
0.04 0.08 0.11 0.15 0.18 0.20
0.31 0.32 0.33 0.34 0.35 0.36
We can now draw the best fitting straight line
10log x
10log y
Take 2 points on the line
(0.04, 0.31) and (0.18, 0.35)
0.35 0.310.29
0.18 0.04m
To find c, use: y mx c
0.35 0.29 0.18 c
0.30c
0.29m 0.30c
Recall our initial suggestionny ax
Taking logs of both sides log log ny ax
log log log ny a x
log log logy a n x
0.30c 0.29m
logY nX a
n = 0.29
10log 0.3a 0.310 1.995...a
a = 2 (1 dp)
0.32y xOur model is approximately
Putting it into practice
1. From the Graph find the gradient
2. From the Graph find or calculate the y-intercept
Make sure the graph shows the origin, if reading it directly
3. Take logs of both sides of suggested function
4. Arrange into form of a straight line
5. Compare gradients and y-intercept
Qu. 1 Assumeny ax Express equation in logarithmic form
log log ny ax
log log log ny a x
log log logy a n x
From the graph:
m = 0.70.6 0.2
0.6666....0.6 0
m
c = 0.2
0.71.6y xRelation between x and y is:
Find the relation between x and y
ny ax
n = 0.7 10log 0.2a 0.210a 1.584...a
Qu. 2 Assumeny ax Express equation in logarithmic form
log log ny ax
log log log ny a x
log log logy a n x
From the graph:
m = -0.70.4 0
0.6666....0 0.6
m
c = 0.4
0.72.5y xRelation between x and y is:
Find the relation between x and y
ny ax
n = -0.7 10log 0.4a 0.410a 2.511...a
Qu. 3
ny ax
When log10 y is plotted against log10 x, a best fitting straight line
log log ny ax
log log log ny a x
log log logy a n x
Given
m = 2 c = -0.8
20.2y xRelation between x and y is:
has gradient 2 and passes through the point (0.6, 0.4)
ny ax
n = 2 10log 0.8a 0.810a 0.158...a
Fit this data to the model
Using
y mx c 0.4 2 0.6 c
Qu. 4
ny ax
When log10 y is plotted against log10 x, a best fitting straight line
log log ny ax
log log log ny a x
log log logy a n x
Given
m = -1 c = 1.1
112.6y xRelation between x and y is:
has gradient -1 and passes through the point (0.9, 0.2)
ny ax
n = -1 10log 1.1a 1.110a 12.589...a
Fit this data to the model
Using
y mx c 0.2 1 0.9 c
Qu. 5 Assumexy ab Express equation in logarithmic form
log log xy ab
log log log xy a b
log log logy a x b
From the graph:
m = 0.00250.3 0.15
0.002560 0
m
c = 0.15
1.4 1.01x
y Relation between x and y is:
Find the relation between x and y
xy ab
10log 0.15a 1.412...a 10log 0.0025b 1.005...b
Qu. 6 Assumexy ab Express equation in logarithmic form
log log xy ab
log log log xy a b
log log logy a x b
From the graph:
m = -0.0150.6 0
0.0150 40
m
c = 0.6
4.0 0.97x
y Relation between x and y is:
Find the relation between x and y
xy ab
10log 0.6a 3.981...a 10log 0.015b 0.966...b
2002 Paper I
11. The graph illustrates the law ny kx
If the straight line passes through A(0.5, 0) and B(0, 1). Find the values of k and n. ( 4 )
logeP p
2000 Paper II
B11. The results of an experiment give rise to the graph shown.
a) Write down the equation of the line in terms of P and Q. ( 2 )
It is given that
and logeQ q
b) Show that p and q satisfy a relationshipof the form bp aq
stating the values of a and b. ( 4 )
THE END
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