1. Using Logs to Linearise Curves
How to linearise y = axb
Demo for Swine Flu CW
Link
How to linearise y = abx
Link

2. Linearising a power equation using logs y = axb Swine Flu2 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

3. Linearising a power equation using logs y = axb
log y = log (axb)
log y = log a + logxb
log y = log a + blogx
log y = blogx + log a
Taking logs of both sides
log(ab) = log(a) + log(b)
log(ax) = xlog(a) Y m X c
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

4. From the graph gradient y intercept m = 1.3962 c = 0.5485 x 2 4 6 8 1012 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
gradient
y intercept
m =
c =

5. y = axb Y m X c
log y = blogx + log a
Y = X
gradient = m = = b
y intercept = c = = log a
Forwards and backwards
a log it 
 10 it = a
a = = 3.54

6. Check if the answer is consistent with the table
Using the equation
y = 3.54x
If x = 5.5 find y
y = 3.54×
= 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

7. Using the equation
y = 3.54x
If y = 100 find x
100 = 3.54x1.3962
log100 = log(3.54x1.3962)
= log(3.54)+log(x1.3962)
= log(3.54) log(x)
log both sides
log(ab) = log(a) + log(b)
log(ax) = xlog(a)

8. Forwards and backwards
log 100= log(3.54) log(x)
Forwards and backwards
x log it  ×  +log3.54 = log 100
log100  –log 3.54  ÷  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 = find x
x = which is consistent with the table

9. Using logs to Linearise the Datax 1 2 3 4 5 6 7 8 9 10 y
111 98 87 77 69 61 54 47 42 38
The equation is y = abx
This graph is NOT linear

10. Using logs to Linearise the Datax 1 2 3 4 5 6 7 8 9 10 y
111 98 87 77 69 61 54 47 42 38
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
Take logs of both sides

11. 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

12. Plot x values on the x axis and logy values on the y axis x 1 2 3 4 56 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
Plot x values on
the x axis and
logy values on
the y axis

13. The equation of the line is
y = x Y m X c
log y = logb x + loga
Matching up : y = mx + c
gradient = log b =
C = log a =

14. gradient = log b =
To find b do forwards and back
b  log it = –0.0522
Backwards
–  10 it  b
b = 10– =

15. y intercept = log a =
To find a do forwards and back
a  log it =
Backwards
 10 it  a
a = = 125.1

16. The exponential equation is y = abx
y = 125.1×0.887x

17. Check if the answer is consistent with the table
Using the equation
y = 125.1×0.887x
If x = 5.5 find y
y = 125.1×
= 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

18. 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 rule log(AB) = logA + logB
Using the drop down
infront rule

19. Forwards and backwards
log 65 = log(125.1)+xlog(0.887)
Forwards and backwards
x ×log0.887  +log = log65
log65  –log  ÷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