Table of Contents

• Prequiz...
• The Power Function
• 1: Introduction
• 2: What are scaling relationships?
• 3: The power function
• 4: The shape of the power function
• 5: The conversion factor "a"
• 6: The power factor "b"
• Log Transformations
• 7: Introduction
• 8: What are logarithms again?
• 9: Graphing on a log scale
• 10: Visualizing the data
• 11: Getting to a straight line
• 12: The key parameter is "b"
• The value of "b"
• 13: Introduction
• 14: A geometric approach
• 15: Scaling surface area of cubes
• 16: Scaling surface area of cubes
• 17: A "2/3 scaling law"?
• Confronting ideas with data
• 18: Introduction
• 19: An unexpected result
• 20: A "3/4 scaling law"?
• 21: Internal transport systems
• 22: Branching and supply networks
• 23: Deriving the "3/4 scaling law"
• 24: So what's the real value of "b"?
• Review
• 25: Review of Concepts

The key scaling parameter is still "b"

So, a brief reminder:

Our original power function: Y = aX^b

And the log-transformed version: log(Y) = b*log(X) + log(a)

Now, the "b" and "a" play a different role in affecting the shape of the function. In this linear form, the "b" is the slope of the line, and log(a) is where the line crosses the y-axis (the y-intercept). But what about their biological meaning? That's what we really care about, after all.

In the orginal power function, "a" was the a conversion factor that didn't mean very much biologically and "b" was the scaling factor that related how the relationship between size and metabolic rate changed at different sizes (so "b" was the parameter we were really interested in). If you need a reminder about this, you may want to take a few minutes and review the first section of this module.

So, what role do the parameters "a" and "b" play in the log-transformed function? Luckily, it turns out they mean the same thing. The parameter "a" is a conversion factor (that we aren't really interested in) - and "b" is the value that relates how size and metabolic rate change across different sizes. The value of "b" is equivalent whether you are using the power function or the log-transformed original, and therefore should be interpreted the same way. Just as we determined earlier, "b" will be a value between 0 and 1.

It may be confusing that with the original power function, when b=1 (and the relationship was therefore linear), it meant that scaling was consistent across all size classes (and the relationship was isometric, and therefore boring), but that isn't true here even though the log-transformed function is always linear. That's because even though the function is linear, the relationship isn't, because the axes are now logarithmic (so the change in rate comes from the values being scaled logarithmically). OK?

Now that we are comfortable looking at our log-transformed graphs and understand the interpretation of our most important parameter, "b" - we are going to spend some time thinking about the exact value of little "b". What do we think that value should be? What is the actual value of little "b" ? And what does it all tell us about the relationship between size and metabolic rate?