Table of Contents

• Prequiz...
• Everything you always wanted to know about logs (but were too bored to ask)
• 1: When size matters
• 2: There must be a better way!
• 3: Special cases
• 4: The log is the power!
• 5: What about messy numbers?
• 6: And SMALL messy numbers?
• 7: Your turn with messy numbers
• 8: Antilogs
• So what are they good for?
• 9: Measuring earthquakes
• 10: Measuring acidity
• 11: Differences in acidity
• Review
• 12: Summary

Learning Outcomes:

After completing this module you should be able to:

• Work with very large and very small numbers by converting between log and linear scales, using logarithms (to the base 10) and anti-logarithms.
• Use the log scale of hydrogen ion concentration (pH) to determine the acidity of a solution.
• Determine the hydrogen ion concentration of a solution from its pH.

When size matters

As you may have noticed, scientists like to measure things, not just anything they can get their hands on, but also lots of things that are too big to get their hands on, and lots of things that are too small. For example:

too big	too small
rainforests planets solar systems galaxies universes	grains of pollen blood cells E. coli enzymes carbon atoms electrons

 

What ends up happening is that we need a lot of zeros to express these sizes. For example:

The number of stars in our galaxy = 100,000,000,000 (roughly)

The distance from here to the edge of the universe =
94,608,000,000,000,000,000,000 km (roughly!)

The problem is just as bad when we talk about small things:

The diameter of a red blood cell = 0.000007 m

The mass of a carbon atom = 0.0000000000000000000000199 grams

Obviously these are numbers that are hard to deal with. A red blood cell with a diameter of 0.00007 m instead of 0.000007 m would be 10 times too big to fit through a capillary, yet it's hard to tell the numbers apart by looking at them – and hard to remember how many zeros there are supposed to be in the first place!

Of course one solution to this problem is to use specialised units. For example, we usually talk about the size of the universe in light years, while we express the size of a red blood cell in micrometres. However, these different units make it hard to compare the sizes of things.