We will sometimes write instead of . Thus, for example

We will also write and for and respectively.

Proof: Let be a partition of , and let

be the partition of obtained by multiplying the points of by .

and

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

whenever , and . Use this result to show that for and

(5.62) |

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

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:

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.

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

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

Also

so

and it follows from this that

Hence

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 .

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.

The graph of the logarithm function is shown below.

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.

Now

and

i.e.,

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.