1.5.6 Logarithms

In Section 1.4 we introduced additive and multiplicative functions by means of the functional equations

`ftext[(]a+btext[)]=ftext[(]atext[)]+ftext[(]btext[)]`   and   `ftext[(]abtext[)]=ftext[(]atext[)]ftext[(]btext[)]`,

respectively. That is, an additive function turns any sum of numbers in its domain into another sum, and a multiplicative function turns any product into another product.

Now we ask: What functions turn sums into products, and what functions turn products into sums? Whatever the answers are, you should recognize now that the verbal descriptions of these two classes of functions are inverses of each other. More specifically, if a given function is in the first of these classes, its inverse (if it's a function) will be in the second class — and vice versa.

From your previous study of algebra, you should know at least one type of function that turns products into sums — that's the whole point of logarithms. You also know properties of logarithms, one of which is

`log text[(]A*Btext[)] = log A + log B`,

where `A` and `B` are any numbers that have logarithms. (What numbers have logarithms?) This logarithmic property says that `log` is a function that satisfies the functional equation for turning products into sums.

Perhaps more important for what we will do in the next chapter is the logarithmic property of turning exponents into factors:

`logtext[(]A^Btext[)]=B* log A`.

But what's a logarithm?

There are many different `log` functions, one for each allowable base. We recall the definition of logarithm base `b`, more or less as it appeared in your algebra or precalculus book: