Would this discussion of logarithms make them easier to understand?
There is a sense in which logarithms demonstrate that addition is structually equivalent to multiplication. Imagine a mirror that reflects multiplication into a logarithm world. When you multiply, a x b, and a and b are reflected as log(a) and log(b). The multiplication is reflected as addition. The result of the multiplication is reflected. Log(a x b) = log(a) + log(b).
People have trouble remembering that log(1)=0. Think of it this way. 1 is the mutiplicative identity. 1 x a = a for all a. The log reflection maps 1 to the addiitive identity in log world. 0 is the additive identity. a + 0 = 1 for all a. So log(1) = 0. The identity in one world translates to the identity in the other one.
1/a is the multiplicative inverse of a, since a x (1/a) = 1, the multiplicative identity. Similarly, -a is the additive inverse, so log(1/a) = -log(a).
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6 Answers
You’re better off teaching logarithms with actual use cases. What you wrote is perfectly fine as a theoretical treatment.
But if you want people to understand and apply what they learn, it needs to be relevant and useful, with examples of how it solves a real problem for them.
I looked at your approach, which I mostly understood, but as I was reading it, I asked myself – what would I do with this knowledge? And I came up with an answer: “not relevant for anything I do”.
Relevance matters.
I agree there should be specific examples to show how things work. I was just trying to come up with an overview, which I have never seen being used.
You could re-invent the slide rule.
A slide rule may be useful for demonstrating logarithms. You could constuct one which had powers of 10 at equal distances along the slide rule and then show how 10^^2×10^^3 = 10^^5.
No you lost me at log(a).
I learned logs by playing with my graphics calculator, and news talking about the Richter Scale for Earthquakes.
Objects in the mirror are closer than they appear.
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