TPC: How To Graph Large Numbers: Semi-Log Graphs.Due _________

In your previous blog, you wrote about the fast spread of information on the internet using unrealistically conservative models: y = 3^x and y=10^x. You let values of x become as large as 15; at that point the numbers became so large that you couldn't graph them on the graphing paper.  Some folks scaled their y axis so that they could include the larger numbers (great job!), but then it was impossible to see the level of growth for the first several hours.  You watched Sal Khan name and discuss these exponential functions in his video posted in my last blog.

So basically, on your graphs, when you show what happens for the larger values of x, you can't see what's going on for the smaller values of x (like the y intercept or the y values when x is 1 or 2 or 3....).  If you have your axis settings so that you can see what happens for the smaller values of x, you can't even include the coordinates for the larger values of x on your graph.  These functions simply increase too rapidly to show everything clearly.

What to do?

You're not the first set of folks to be concerned about this and to be frustrated with the graphs.  So we have a different way of graphing.  We use log graphs.  I have log graph paper copied for you already to use.  You'll be able to show all values of the exponential functions above on these graphs.  It's actually called "semi-log" graph paper because one side is spaced constantly (like you are used to) and the second side is spaced differently -- explained below.

When you look at this paper, notice the spacing of the vertical lines.  These, obviously, measure the "x" values.  The spacing is even; the physical distance on the graph is a constant one unit.

What you'll notice that is different is the spacing on the horizontal lines, the "y" values.  Start with the darkened horizontal lines that are evenly spaced.  You'll see the first one, about an inch and a half up the page labeled as 10^1 (sometimes it's simply labeled as "1" in semi-log paper) then the second darker line, about 3 inches up the page labeled 10^2 and so forth.  These integers represent powers of 10.  What is 10^0?   I've marked some of these values for you. Label the ones I did not label.

Now focus on the horizontal lines between 10^0=1 and 10^1=10. You'll see that as you move your eyes up the page, these lines become closer together.  The first line represents the value 1, the second 2, and so forth. You're counting by ones.

Now focus on the lines between 10^1 and 10^2.  These lines also become closer together as you move up the page.  The distance between each line is now 10 units.  So you can count by tens between the "1" (which represents 10^1) and the "2" (which represents 10^2). Count the lines: 20,30,40,50... up to the darkened line which represents 100.  I've labeled some for you.

The pattern continues. Between the 10^2 and the 10^3, each line represents an increase of 100.  I've labeled some of these for you. Between the 10^2 and the 10^3, you count by hundreds.



To the right is a graph of three points. This graph paper uses a "10" instead of what our paper labels a "10^1."  It also uses "100" when our graph paper labels a "10^2."   The coordinates of these three points to the right are:
(1,10), (2,40) and (3,1). Match the coordinates to each point on the graph to the right.







To the right is a second image of points on a log graph. In your blog, identify the coordinates of those three points. If you are having trouble, the answers are given 11:19 in the YouTube video linked below. If you think you've got it, then do check your answers by watching the video linked below.







Continue to label the magnitudes represented by each horizontal line on your graph paper.  Here's a YouTube that will help. Watch that video now. It's about 12 minutes.  At 7 minutes, the measurement on the graph will start at "1" like we are in our graph.  This video will help you verify that you've labeled your horizontal lines properly (notice the paper I've given you is only different in how it starts.)

Then refer to your charts for y = 3^x and y = 10^x.  Plot those points on the log paper graph that I've given you. You'll be able to graph all the points for your charts.  You'll also see an interesting and simple relationship.  See if that relationship holds for another exponential function. (You can chose any base, but y = 2^x works just fine, too.) So you've now graphed THREE exponential functions on log paper.

For your blog, you will (a) turn in your three graphs on the same semi-log paper.  (b)  Write in your blog your observations about what these exponential functions graphed like on log paper. (c) Find something new you learned about this kind of graphing that was in the youtube video.  There's lots of gems in this video; pull out at least three ideas from the video to put in your blog. One could be the rhyme, another could be the proportionality issue, another could be how decimals are represented....your choice.  You also could discuss why you think these exponential functions graphed as they did on semi-log paper.