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

Linearising a power equation using logs y = axb Swine Flu 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 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

From the graph gradient y intercept m = 1.3962 c = 0.5485 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 gradient y intercept m = 1.3962 c = 0.5485

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

Check if the answer is consistent with the table 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(3.54)+log(x1.3962) = log(3.54)+1.3962log(x) log both sides log(ab) = log(a) + log(b) log(ax) = xlog(a)

Forwards and backwards 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 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

Using logs to Linearise the Data x 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

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

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 Plot x values on the x axis and logy values on the y axis

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

Check if the answer is consistent with the table Using the equation y = 125.1×0.887x If x = 5.5 find y y = 125.1×0.8875.5 = 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) = 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

Forwards and backwards 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