Transcript Using Logs to Linearise Revision qu.s

Using Logs to Linearise Curves
How to linearise y = axb
Demo for Swine Flu CW
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How to linearise
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y = abx
y = axb
Linearising a power equation using logs
Swine Flu
x
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y
9.5
24
43
64
87.9
113
141
170
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232
265
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This graph is NOT linear
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Linearising a power equation using logs
y = axb
y = axb
log y = log (axb)
Taking logs of both sides
log y = log a + logxb
log(ab) = log(a) + log(b)
log y = log a + blogx
Y
m X
c
log y = blogx + log a
log(ax) = xlog(a)
This is of the form y = mx + c.
gradient = b
y intercept = log a
So make a new table of values where
Y = log y and
X = log x
x
2
4
6
8
10
12
14
16
18
20
22
y
9.5
24
43
64
87.9
113
141
170
200
232
265
x=log x 0.30 0.60 0.78 0.90 1.00 1.08 1.15 1.20 1.26 1.30 1.34
y=log y 0.98 1.38 1.63 1.81 1.94 2.05 2.15 2.23 2.30 2.37 2.42
Graph of log y against log x
3
From the graph
m = 1.3962
gradient
y intercept
c = 0.5485
2.5
y = 1.3962x + 0.5485
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log y
1.5
1
0.5
0
-0.5
0
0.5
1
log x
1.5
2
y = axb
Y
m X
c
log y = blogx + log a
Y = 1.3962X + 0.5485
gradient = m = 1.3962 = b
y intercept = c = 0.5485 = log a
Forwards and backwards
a log it  0.5485
0.5485  10 it = a
a = 10 0.5485 = 3.54
Using the equation
y = 3.54x 1.3962
If x = 5.5 find y
y = 3.54×5.5 1.3962
= 38.2
Check if the answer is consistent with the table
x
2
4
6
8
10
12
14
16
18
20
22
y
9.5
24
43
64
87.9
113
141
170
200
232
265
x = 5.5 find y
y = 38.2 which is consistent with the table
Using the equation
y = 3.54x 1.3962
If y = 100 find x
100 = 3.54x1.3962
log100 = log(3.54x1.3962)
log both sides
= log(3.54)+log(x1.3962)
log(ab) = log(a) + log(b)
= log(3.54)+1.3962log(x)
log(ax) = xlog(a)
log 100= log(3.54)+1.3962log(x)
Forwards and backwards
x log it  ×1.3962  +log3.54 = log 100
log100  –log 3.54  ÷1.3962  10 it = x
x = 10.97
Check if the answer is consistent with the table
x
2
4
6
8
10
12
14
16
18
20
22
y
9.5
24
43
64
87.9
113
141
170
200
232
265
y = 100 find x
x = 10.97 which is consistent with the table
Using logs to Linearise the Data
x
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y
111
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87
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38
The equation is y = abx
140
This graph is NOT linear
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Using logs to Linearise the Data
x
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y
111
98
87
77
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61
54
47
42
38
The equation is y = abx
log y = log(abx)
Take logs of both sides
log y = log a + logbx
Using the addition rule log(AB) = logA + logB
log y = log a + (xlogb)
Using the drop down infront rule
log y = (logb) x + loga
Rearranging to match with y = mx + c
log y = (logb)x + loga
Rearranging to match with y=mx + c
Matching up :
Y axis = log y
gradient =m = logb
x axis = x
C = log a
So make a new table of values
x=x
Y = logy
x
1
2
3
4
5
6
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8
9
10
y
111
98
87
77
69
61
54
47
42
38
logy 2.05 1.99 1.94 1.89 1.84 1.79 1.73 1.67 1.62 1.58
Plot x values on
2.50
2.00
the y axis
logy
the x axis and
logy values on
y = -0.0522x + 2.0967
1.50
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0.50
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The equation of the line is
y = -0.0522x + 2.0973
Y
m X
c
log y = logb x + loga
Matching up : y = mx + c
gradient = log b = -0.0522
C = log a = 2.0973
gradient = log b = -0.0522
To find b do forwards and back
b  log it = –0.0522
Backwards
–0.0522  10 it  b
b = 10–0.0522 = 0.8867
y intercept = log a = 2.0973
To find a do forwards and back
a  log it = 2.0973
Backwards
2.0973  10 it  a
a = 102.0973 = 125.1
The exponential equation is y = abx
y = 125.1×0.887x
Using the equation
y = 125.1×0.887x
If x = 5.5 find y
y = 125.1×0.8875.5
= 64.7
Check if the answer is consistent with the table
x
1
2
3
4
5
6
7
8
9
10
y
111
98
87
77
69
61
54
47
42
38
x = 5.5 find y
y = 64.7 which is consistent with the table
Using the equation
y = 125.1×0.887x
If y = 65 find x
65 = 125.1×0.887x
log65 = log(125.1×0.887x)
Take logs of both sides
= log(125.1)+log(0.887x)
Using the addition rule
log(AB) = logA + logB
= log(125.1)+xlog(0.887)
Using the drop down
infront rule
log 65 = log(125.1)+xlog(0.887)
Forwards and backwards
x ×log0.887  +log 125.1 = log65
log65  –log 125.1  ÷log0.887 = x
x = 5.46
Check if the answer is consistent with the table
x
1
2
3
4
5
6
7
8
9
10
y
111
98
87
77
69
61
54
47
42
38
y = 65 find x
x = 5.46 which is consistent with the table