log-log and semi-log plots: why and how to teach

[daijobu]

I'd like to introduce my kids to plotting on log-log and semi-log graphs, but frankly I hardly understand why this is necessary myself.  

 

Is there a good teaching resource on this topic and can someone explain why we even do this at all?  

[kiana]

Usually when data has an exponential relationship -- if you are trying to plot (for example) 10^x, a standard scale will have it off the paper by the time you get to x=2 and through the ceiling by the time you get to x=4. 

 

But I don't have a great teaching resource because I've never taught it. I've just used it for displaying data in my own work. 

[Arcadia]

I used that for Physics in middle school rather than for Maths in the late 80s. Most likely I was taught in both subjects to plot on log-log graph paper, as well as calculating the log of the variables and plotting on normal graph paper.

 

Example and explanation in Physics and Maths

 

http://www.physics.pomona.edu/sixideas/labs/lrm/lr05.pdf

 

http://www.tech.plym.ac.uk/maths/resources/PDFLaTeX/loglog.pdf

 

If you want to let your kids have fun trying, you can print the graph papers in link

http://www.physics.utah.edu/~gernot/elem_labs/Graph_paper.htm

 

ETA:

Mathematica and Mathlab does log-log plots. I had used both at university.

[EmilyGF]

I don't know when my hubby doesn't use a log-log or semi-log plot. He's an astronomer and his numbers are huge and very widely.

[daijobu]

I can't find anything in my physics textbook (Serway and Jewett, Physics for Scientists and Engineers).  Is it typically taught in precalculus or some other math class?

[daijobu]

Okay, here's my plan in case anyone is following.  I've bought 100 regular dice.  We start with one die and pretend it's a cancer cell.  Each roll represents a unit of time.  If the roll is a 2, then we double the number of dice, that is, we now have 2 dice.  That represents a cell division.  Then we continue by rolling the two dice, until another 2 is rolled that represents another cell division.  Continue on, keeping track of the number of cancer cells/dice at each moment in time and graph it and put the data into a table.  

 

In the table have 3 columns: time, number of cells and log(number of cells) base 10.  Now graph log(# cells) versus time, then by eyeballing it with a straight edge, draw a "best fit" line.  Compute the approximate slope, m,  (again by eyeballing) of this line.  Now y=log(#cells) = mx.  Or 10^mx = # cells.  Now go back to the exponential plot and plot y = 10^mx and see how well it fits the original data.

 

Repeat these calculations using Excel.  

 

I tried it myself and got m=7 and when I plotted y=10^7x, my model rose more steeply than my actual data, which I thought made sense, since shouldn't I expect m=6?  

 

What do you think of this approach?  

[regentrude]

I can't find anything in my physics textbook (Serway and Jewett, Physics for Scientists and Engineers).  Is it typically taught in precalculus or some other math class?

 

It is typically never taught in high school, except maybe in a very good physics lab. Most of our physics majors students  are completely unfamiliar with the concept and have to be taught in lab or project based class at college. But then, there really is not much to it - you just have to show it to the student. The real difficulty is taking your actual data and deciding when a log plot is appropriate; this usually comes with experience.

 

[regentrude]

Okay, here's my plan in case anyone is following.  I've bought 100 regular dice.  We start with one die and pretend it's a cancer cell.  Each roll represents a unit of time.  If the roll is a 2, then we double the number of dice, that is, we now have 2 dice.  That represents a cell division.  Then we continue by rolling the two dice, until another 2 is rolled that represents another cell division.  Continue on, keeping track of the number of cancer cells/dice at each moment in time and graph it and put the data into a table.  

 

In the table have 3 columns: time, number of cells and log(number of cells) base 10.  Now graph log(# cells) versus time, then by eyeballing it with a straight edge, draw a "best fit" line.  Compute the approximate slope, m,  (again by eyeballing) of this line.  Now y=log(#cells) = mx.  Or 10^mx = # cells.  Now go back to the exponential plot and plot y = 10^mx and see how well it fits the original data.

 

Repeat these calculations using Excel.  

 

I tried it myself and got m=7 and when I plotted y=10^7x, my model rose more steeply than my actual data, which I thought made sense, since shouldn't I expect m=6?  

 

What do you think of this approach?  

 

I find this an extremely complicated setup whose mathematical difficulty in calculating the probabilities to obtain a theoretical result for comparison obscures the learning objective of the logarithmic plotting.

May it's because my brain is not quite awake (no coffee), but I admit I have a hard time figuring out why I should expect this to follow a 10^6x curve.

 

If it is about teaching logarithmic graphing, I would use a much simpler relationship first which the student can easily understand from theory. Find something that has a relationship that involves a clean, simple exponent, such as a square root. Pendulum is  a good example; the period is proportional to the square root for the length of the string. Plot linear- hard to see that the curve is square root. Plot in log - straight line with slope 1/2.

 

[Arcadia]

I can't find anything in my physics textbook (Serway and Jewett, Physics for Scientists and Engineers).

Mine was Nelson & Parker as well as Halliday & Resnick whatever edition was around in the 80s. One of the areas it was covered is in link below (4 page).

http://users.df.uba.ar/sgil/physics_paper_doc/papers_phys/e&m/rlc_inductance_94.pdf

 

For an experiment involving log plotting, the easiet would probably be the pendulum experiment. Something like the link below

http://www.ysu.edu/physics/tnoder/F06-PHYS2610L/Sample-LabReport.pdf

 

ETA:

Another log log example, page 7 of 14

http://ocw.mit.edu/courses/materials-science-and-engineering/3-11-mechanics-of-materials-fall-1999/modules/ss.pdf