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

What are logarithms again?

First, let's all just remind ourselves what a logarithm is (those who remember their logs well can skip to the next page!). The logarithm of a number is the exponent you raise above 10 to get that number. This is best seen by examples.

log(100) = 2 (why? because 10² = 100)

log(10,000) = 4 (why? because 10⁴ = 10,000)

An easy rule that works for multiples of 10 is that the log is equal to the number of zeros trailing the one (go ahead and count the zeros!):

log (10,000,000) = 7

log (1,000,000,000,000) = 12

These multiples of 10 are always easy, but you can take the log of any number (in this case, we suggest you use your calculator- just type in the number, then hit the "log" button).

log(3,462) = 3.539327 (why? because 10^3.539327 = 3,462)

Logs can also be figured for numbers less than one. When a number is a fraction (less than one), then the log is always negative.

log(0.01) = -2 (why? because 10^-2 = 0.01)

Why does this work? Because 10^-2 is the same as 1/10², which equals 1/100, which equals 0.01!

log(0.0001) = -4 (why? because 10^-4 = 1/10⁴ = 1/10,000 = 0.0001)

An easy rule that works for decimals that are multiples of 0.1 is that the log is equal to the number of zeros trailing the decimal plus the "1" (go ahead and count those zeros again!):

log (0.0000001) = -7

log (0.000000000001) = -12

Again, these multiples of 0.1 are always easy, but you can take the log of any positive decimal (but again, we suggest using your calculator!):

log(0.3462) = -0.4607 (why? because 10^-0.4607 = 0.3462)

What happens if you take the log of zero? Well, how many times would you have to multiply 10 by itself to get zero? Well, if you think about it, there is no amount of times you can multiply 10 by itself and get zero! That means that the quantity log (0) is undefined (go ahead and try it on your calculator!). This is true also for negative numbers. Since you are always starting with a positive number (10) - and always multiplying by a positive number (10), you're just never going to get a negative number. So, give up your dreams of taking the log of a negative number - it just isn't going to work!

Oh, one more thing about logs. You probably remember that you can take the log with bases other than 10 (so log₂ is the exponent you would raise above 2 to get a particular number). But in this scaling module, we are always refering to logarithms of base 10 (hooray - because those are much easier to think about!).

Want more background and practice with logs? Visit the Logs and pH MathBench module.