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# 5.4 Logarithms.

5.57   Notation (, .) Let be a bounded function from the interval to . We will denote the area of by . Thus

We will sometimes write instead of . Thus, for example

We will also write and for and respectively.

5.58   Lemma. 5.2 Let and be real numbers such that and . Then

Proof: Let be a partition of , and let

be the partition of obtained by multiplying the points of by .
Then
 (5.59)

and
 (5.60)

We know that

Hence by (5.59) and (5.60) we have

for every partition of . It follows from this and the last statement of theorem 5.40 that

5.61   Exercise. A From lemma 5.58 we see that

whenever , and . Use this result to show that for and
 (5.62)

5.63   Definition (.) We will define a function by

By exercise 5.61A we have
 (5.64)

In this section we will extend the domain of to all of in such a way that (5.64) holds for all .

5.65   Theorem. Let be real numbers such that , and let be a bounded function from to . Then
 (5.66)

Proof: We want to show

Since and the sets and are almost disjoint, this conclusion follows from our assumption about additivity of area for almost disjoint sets.

I now want to extend the definition of to cases where may be less than . I want equation (5.66) to continue to hold in all cases. If in (5.66), we get

i.e.,

Thus we make the following definition:

5.67   Definition. Let be real numbers with and let be a bounded function from to . Then we define

5.68   Theorem. Let be real numbers and let be a bounded non-negative real valued function whose domain contains an interval containing , and . Then

Proof: We need to consider the six possible orderings for and . If we already know the result. Suppose . Then and hence , i.e., . The remaining four cases are left as an exercise.

5.69   Exercise. Prove the remaining four cases of theorem 5.68.

5.70   Definition (Logarithm.) If is any positive number, we define the logarithm of by

5.71   Theorem (Properties of Logarithms.) For all and all we have
 (5.72) (5.73)

Proof: Let . From lemma 5.58 we know that if then

 (5.74)

If we get

so equation (5.74) holds in all cases. Let be arbitrary elements in . Then

Also

so

and it follows from this that

Hence

5.75   Lemma. For all , .

Proof: The proof is by induction on . For the lemma is clear. Suppose now that the lemma holds for some , i.e., suppose that . Then

The lemma now follows by induction.

If then and

Thus equation (5.72) holds whenever . If and , then

so

Thus (5.72) holds for all .

5.76   Theorem. Let and be numbers such that . Then

Proof: By lemma 5.58

Logarithms were first introduced by John Napier (1550-1632) in 1614. Napier made up the word logarithm from Greek roots meaning ratio number, and he spent about twenty years making tables of them. As far as I have been able to find out, the earliest use of for logarithms was by Irving Stringham in 1893[15, vol 2, page 107]. The notation is probably more common among mathematicians than , but since calculators almost always calculate our function with a key called ln'', and calculate something else with a key called log'', I have adopted the ln'' notation. (Napier did not use any abbreviation for logarithm.) Logarithms were seen as an important computational device for reducing multiplications to additions. The first explicit notice of the fact that logarithms are the same as areas of hyperbolic segments was made in 1649 by Alfons Anton de Sarasa (1618-1667), and this observation increased interest in the problem of calculating areas of hyperbolic segments.

5.77   Entertainment (Calculate .) Using any computer or calculator, compute accurate to 10 decimal places. You should not make use of any special functions, e.g., it is not fair to use the ln" key on your calculator. There are better polygonal approximations to than the ones we have discussed.

The graph of the logarithm function is shown below.

We know that and it is clear that is strictly increasing.
If , then

From the fact that for all , it is clear that takes on arbitrarily large positive and negative values, but the function increases very slowly. Let

be the regular partition of into three subintervals.
Then

Now

and

i.e.,
 (5.78)

There is a unique number such that . The uniqueness is clear because is strictly increasing.

The existence of such a number was taken as obvious before the nineteenth century. Later we will introduce the intermediate value property which will allow us to prove that such a number exists. For the time being, we will behave like eighteenth century mathematicians, and just assert that such a number exists.

5.79   Definition (.) We denote the unique number in whose logarithm is by .

5.80   Exercise. A Prove that . (We already know .)

5.81   Entertainment (Calculate .) Using any computing power you have, calculate as accurately as you can,e.g., as a start, find the first digit after the decimal point.

Next: 5.5 Brouncker's Formula For Up: 5. Area Previous: 5.3 Monotonic Functions   Index
Ray Mayer 2007-09-07