Logarithms are suitable for projects.

Given the memory pegs for the common logs and natural logs find the rule that connects them.

This is a good project that requires experimentation. From it one finds the change of base rule for logarithms.

A follow on project is to try to prove the rule by using fractional powers. Fractional powers themselves can be a project. Find a rule to define what fractional power means.

10^{m/n} = x

means what?

find an x so that

10^m = x^n

10^1 = x^2

we search for the x that is closest.

For logs we can use this to find logs to base 2, 3, 10.

We want to find the base 10 log of 17, which is prime. One approach is as follows.

2^4 = 16,

2*3*3 = 18

If we have the base 10 logs of 2 and 3 already, we can sandwich the log of 17 between them.

Why is this valid? Are we assuming something? Can we prove it? How?

How were logarithm tables constructed c. 1600 before calculus?

What tricks or computation aids can be used to simplify the process?

We want to reduce the construction of the table to addition as much as possible. How can we do this?

Which numbers are easy and which hard to find the logarithm? To add precision to the answer?

Do we need the rules of exponents first? Or can we be led to rules of exponents by experimentation with this task?

How do we use the log tables we can build to do multiplication? Do we need to make approximations? In constructing the log tables? In the calculation? How do we construct bounds on the answer? On the errors? Is the answer defined?

If we have applications to multiplication of numbers that are already defined, can we use that to define logarithms? Given the rules we will use for logs?

Is it easier to use natural logs for multiplying two numbers? Common logs, i.e. base 10? Base two?

Are base 2 log tables the easiest to build? How do we go from those to base 10? Who figured this out first? What reasoning did they use? Was that a proof? When was the first proof?

Can we use letters to stand for numbers for definitions? For proofs? How far can we go explaining what we do without using letters? Try this as a project. Then try using letters to explain it.

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## About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition.
President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.