## Reflection: Intrinsic Motivation Logarithms - Napier's Wonderful Invention! - Section 4: Which is Easier?

When I began reflecting on my teaching of logarithms, I realized that I couldn't understand the motivation for them.  They seemed so strange, that it was difficult to imagine the circumstances that would lead someone to discover or invent them.  What problem did they solve for that person, and where does this strange name come from?  If my students were faced with the same problem, the same motivation, maybe they would understand and appreciate logarithms more deeply.  I frequently take this approach because I believe that it will help students to connect new concepts to old ones as well as help them to understand how mathematics develops in history.

The motivation for John Napier was the difficulty of carrying out multiplication and division.  To make life easier for mathematicians, he invented decimal notation (hooray!) and then logarithms (huh?).

To help them appreciate the invention, I wanted them to feel how much more difficult division is when doing it by hand.  I also wanted them to see how logarithms could be used to carry out the division.

To make this happen, I had some students divide two relatively prime integers (572 / 95) to seven decimal places.  Yuck!  Most of them don't make it past the second or third decimal place.  Meanwhile, other students are busily finding the difference of the logarithms of these numbers to the base 4.  In the end, they will see very clearly that subtraction is much easier than addition since those students finished quickly and with very little stress.  They will also see that the ratio of 572 to 95 can be obtained from the difference of the two logarithms!  These logarithms are "ratio-numbers", which is exactly what the name meant to John Napier.  My students can understand why this works because of their mastery of exponents, they can appreciate the motivation for the invention when they are denied their calculators, and they can see that logs are still relevant when they use them to solve an exponential equation.

Feeling the Difference
Intrinsic Motivation: Feeling the Difference

# Logarithms - Napier's Wonderful Invention!

Unit 6: Exponents & Logarithms
Lesson 9 of 14

## Big Idea: What turns multiplication into addition and division into subtraction? Logarithms, that what!

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Subject(s):
Math, Exponents and Exponential Functions, logarithms, Algebra 2, master teacher project
50 minutes

### Jacob Nazeck

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