In your head, can you find the average of 2741, 2742 and 2746?

I am hoping that most of you realize that we can find the average by doing the following:

1. Subtract 2740 from each number, giving 1, 2 and 6.
2. Find the average of 1, 2 and 6 = (1+2+6)/3 = 3
3. Add back 2740 to 3, getting an average of 2743.

The purpose of this exercise is to see if it can be used to explain the very useful mathematical concept of isomorphism. Let me know if this makes any sense.

The procedure used in finding the average can be generalized as follows:
1. Perform a transformation to each of the numbers, in this case subtracting 2740.

2. Do a calculation with the transformed numbers. In this case, the operation is the same, finding an average. In general, the operation need not be identical.

3. Perform the reverse of the original transformation. We started by subtracting 2740 from each number. Now we add it back to get the answer.

Now consider the formula log(x*y) = log(x) + log(y). Suppose we have a table of logarithms. We can use it to multiply two numbers x and y by doing the following.
1. Use the table to convert x and y to log(x) and log(y)
2. In the log world, a multiplication is turned into an addition, We calculate log(x)+log(y)
3. Use the table in the opposite direction to determine what number has a log value of log(x)+log(y)

A lot of people have trouble with log(1)=0. Here is a way of thinking about it. 1 is the multiplicative identity: 1*x=x for any x. In the log world, multiplication is replaced by addition. The multiplicative identity must be transformed into the additive identity, which is 0, since 0+x=x for any x.