Table of Contents

• Section 1. How the Vibrio population grows
• 1: What causes cholera?
• 2: Build the basic growth equation
• 3: Draw exponential growth
• Section 2. So why isn't Frank dead already?
• 4: How long until 1 million bacteria?
• 5: All we need now is data
• 6: Looking at data
• 7: Log transform straightens out exponential growth
• 8: Do ALL curves get "straightened" by taking the log?
• 9: Phases of bacterial growth
• Section 3. How can we tell when the population is growing exponentially?
• 10: A test for exponential growth
• 11: And the answer is...
• 12: To recap
• 13: Some more practice
• 14: Extended problem: Exponential growth ends
• Review
• 15: Summary

Do ALL curves get "straightened" by taking the log?

Try it yourself and find out...

So, although log transformation is powerful, it is not ALL-powerful. The only curves that actually get straightened are the ones generated by functions that include "some number to the power of the x variable". Why is that? Remember that:

any population that GROWS OR SHRINKS BY A CONSTANT MULTIPLICATIVE FACTOR will appear as a straight line when log-transformed.

And that's what doubling does -- it multiplies the population by a constant factor (2, that is) over a period of time. But other mathematical functions, such as the ones above, involve some sort of change in which the factor of change is NOT constant. Let's take the parabola, for example: y = x²

x	1	2	3	4	5	6
y	1	4	9	16	25	36
factor		4	2.25	1.78	1.56	1.44

The factor by which y is increasing starts at 4 (4 divided by 1), then drops to 2.25 (9 divided by 4), then 1.78 (16 divided by 9), and so on. Although we might think that a parabola grows "pretty fast", it is in fact "slow" compared to exponential growth that starts with the same factor.

As a general guideline:

• if growth factors are increasing,
  then growth is proceeding faster than exponential,
  and that part of the log-transformed graph will curve up.
• if growth factors are decreasing,
  then growth is proceeding slower than exponential,
  and that part of the log-transformed graph will have a flattening curve.
• Only constant growth factors (exponential growth)
  leads to a straight-line when the graph is log-transformed.