We now have the tools required to properly define the exponential and the logarithm that you know from calculus so well. We start with exponentiation. If is a positive integer, it is obvious to define
It is convenient to define the logarithm first. Let us show that a unique function with the right properties exists, and only then will we call it the logarithm.
Given by the Archimedean property of the real numbers (notice ), there is an such that The intermediate value theorem gives an such that Thus is in the image of As is increasing, for all and so
See Figure 5.6. Mathematicians usually write instead of which is more familiar to calculus students. For all practical purposes, there is only one logarithm: the natural logarithm. See Exercise 5.4.2.
Again, we prove existence of such a function by defining a candidate and proving that it satisfies all the properties. The defined above is invertible. Let be the inverse function of Property i is immediate.
Let us look at property iii. The function is strictly increasing since As is the inverse of it must also be bijective. To find the limits, we use that is strictly increasing and onto For every there is an such that and for all Similarly, for every there is an such that and for all Therefore,
As is continuous, then is a continuous function of Therefore, we would obtain the same result had we taken a sequence of rational numbers approaching and defined
There are other equivalent ways to define the exponential and the logarithm. A common way is to define as the solution to the differential equation See Example 6.3.3, for a sketch of that approach. Yet another approach is to define the exponential function by power series, see Example 6.2.14.
We proved the uniqueness of the functions and from just the properties and the equivalent condition for the exponential Existence also follows from just these properties. Alternatively, uniqueness also follows from the laws of exponents, see the exercises.
Hint: Take the logarithm. Note: The expression arises in compound interest calculations. It is the amount of money in a bank account after 1 year if 1 dollar was deposited initially at interest and the interest was compounded times during the year. The exponential is the result of continuous compounding.