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How to linearise y = abx
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How to linearise y = axb
Demo for Swine Flu CW
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Using Logs to Linearise Curves
Linearising a power equation using logs y = axb
Swine Flu
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
This graph is NOT linear
Linearising a power equation using logs y = axb
y = axb
log y = log (axb)
log y = log a + logxb
log y = log a + blogx log y = blogx + log a
So make a new table of values where
Y = log y and X = log x
This is of the form y = mx + c.
Taking logs of both sides
log(ab) = log(a) + log(b)
log(ax) = xlog(a)
gradient = b y intercept = log a
Y m X c
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
From the graph gradienty intercept
m = 1.3962c = 0.5485
gradient = m = 1.3962 = b
y intercept = c = 0.5485 = log a
y = axb
log y = blogx + log a
Y = 1.3962X + 0.5485
a = 10 0.5485 = 3.54
0.5485 10 it = a
Forwards and backwards
a log it 0.5485
Y m X c
Using the equation
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 = 5.5 find y
y = 38.2 which is consistent with the table
y = 3.54x 1.3962
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
Using the equation
If y = 100 find x
100 = 3.54x1.3962
log100 = log(3.54x1.3962)
= log(3.54)+log(x1.3962)
= log(3.54)+1.3962log(x)
y = 3.54x 1.3962
log(ab) = log(a) + log(b)
log(ax) = xlog(a)
log both sides
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
x = 10.97 which is consistent with the table
Check if the answer is consistent with the table
y = 100 find 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
Using logs to Linearise the Data
The equation is y = abx
x 1 2 3 4 5 6 7 8 9 10
y 111 98 87 77 69 61 54 47 42 38
0
20
40
60
80
100
120
140
0 2 4 6 8 10 12
This graph is NOT linear
Using logs to Linearise the Data
The equation is y = abx
log y = log(abx)
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
x 1 2 3 4 5 6 7 8 9 10
y 111 98 87 77 69 61 54 47 42 38
Take logs of both sides
So make a new table of values
x = x
Y = logy
Matching up :
Y axis = log y
gradient =m = logb
x axis = x
C = log a
log y = (logb)x + loga Rearranging to match with y=mx + c
Plot x values on
the x axis and
logy values on
the y axis
x 1 2 3 4 5 6 7 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
y = -0.0522x + 2.0967
0.00
0.50
1.00
1.50
2.00
2.50
0 2 4 6 8 10 12
x
logy
y = -0.0522x + 2.0973
The equation of the line is
log y = logb x + loga
gradient = log b = -0.0522
Matching up : y = mx + c
C = log a = 2.0973
Y m X c
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
Backwards2.0973 10 it a
a = 102.0973 = 125.1
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 = 5.5 find y
y = 64.7 which 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
Using the equation
y = 125.1×0.887x
If y = 65 find x
65 = 125.1×0.887x
log65 = log(125.1×0.887x)
= log(125.1)+log(0.887x)
= log(125.1)+xlog(0.887)
Take logs of both sides
Using the addition rulelog(AB) = logA + logB
Using the drop down infront rule