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

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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

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 = 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

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

x = 5.46 which is consistent with the table

Check if the answer is consistent with the table

y = 65 find x

x 1 2 3 4 5 6 7 8 9 10

y 111 98 87 77 69 61 54 47 42 38