MEI Core 2. Logarithms and exponentials. Section 2: Modelling curves using logarithms. Modelling curves of the form y kx

Transcription

1 MEI Core 2 Logarithms ad eoetials Sectio 2: Modellig curves usig logarithms Notes ad Eamles These otes cotai subsectios o: Modellig curves of the form y = k Modellig curves of the form y = ka Modellig curves of the form y k Whe you collect data from a eerimet, you may wat to fid a relatioshi betwee two variables, such as the seed of a movig object at a articular time, or temerature of a object ad its distace from a heat source. You may lot a grah of oe variable agaist aother to hel fid this relatioshi. However, uless the grah is a straight lie, it may be difficult to 2 3 see the relatioshi from the grah. The grahs of y, y etc. look uite similar for 0, ad as the eerimetal data may ot be very accurate, it ca be imossible to tell with ay certaity what would be the best grah to model the data. This is where logarithms ca be very useful. If the relatioshi is of the form y k, the lottig log y agaist log gives a straight lie grah. y k log y log k log y log k log log y log k log log y log log k This is of the form y = m + c, so lottig log y agaist log gives a straight lie with gradiet ad itercet log k. The value of is therefore the gradiet of the grah, ad the value of k is foud by takig the itercet of the grah ad fidig its iverse logarithm (i.e. itercet 10 if you are usig logs to base 10). Eamle 1 The relatioshi betwee two variables ad y is believed to be of the form y k, where k ad are costats. I a eerimet, the followig values of ad y are recorded y MEI, 16/03/09 1/5

2 MEI C2 Logarithms Sectio 2 Notes ad Eamles Verify that the model costats k ad. y k is aroriate ad fid the aroimate values of the Solutio y k Takig logarithms: log y log k log y log k log log y log k log If log y is lotted agaist log, this is the euatio of a straight lie grah with gradiet ad itercet log k. Plot the values of log y agaist log : log y log y Sice the grah is aroimately a straight lie, the relatioshi aroriate model Gradiet = = Itercet log k 0.3 k 10 2 y k is a MEI, 16/03/09 2/5

3 MEI C2 Logarithms Sectio 2 Notes ad Eamles The relatioshi is aroimately y For ractice i roblems similar to this, try the iteractive uestios Euatios from log-log grahs. Modellig curves of the form y ka Similarly, if the relatioshi is of the eoetial form y agaist gives a straight lie grah. y ka, the lottig log y ka log y log ka log y log k log a log y log k log a log y log a log k This is of the form y = m + c, so lottig log y agaist gives a straight lie with gradiet log a ad itercet log k. The value of a is foud by takig the gradiet of the grah ad fidig its gradiet iverse logarithm (i.e. 10 if you are usig logs to base 10), ad the value of k is foud by takig the itercet of the grah ad fidig its iverse itercet logarithm (i.e. 10 if you are usig logs to base 10). Eamle 2 The relatioshi betwee two variables ad is believed to be of the form where ad are costats. ab, I a eerimet, the followig values of ad are recorded Verify that the model ab is aroriate, ad estimate the values of a ad b. Solutio ab Takig logarithms: log log ab log log a logb log log a logb If log is lotted agaist, this is the euatio of a straight lie with gradiet log b ad itercet log a. MEI, 16/03/09 3/5

4 MEI C2 Logarithms Sectio 2 Notes ad Eamles Plot the values of log agaist : log Sice the grah is aroimately a straight lie, the relatioshi aroriate model. ab is a 1.25 Gradiet = log b = b Itercet = log a = 0.5 a The relatioshi is aroimately 1.25/ You do ot eed to remember the details of what to lot ad what to do with the gradiet ad itercet of the grah all you eed to do is to take logs of both sides of the suggested relatioshi ad aly the laws of logarithms to obtai a relatioshi of the form y = m + c, as show above for each of the relatioshis y k ad y ka. Oce you have doe this, you ca see what you eed to lot ad how to fid the values of the costats. MEI, 16/03/09 4/5

5 MEI C2 Logarithms Sectio 2 Notes ad Eamles If the grah is ot a straight lie, the the suggested model is ot a aroriate oe (or erhas the eerimetal results are ot sufficietly accurate). For ractice i roblems like the oe above, try the iteractive uestios Euatios from log- grahs. MEI, 16/03/09 5/5