LOGARITHMS

Before presenting the third stage of numerical structure, a few words on the operation of logarithms might be in order, accompanied by a recommendation not to use this term to designate a specific stage of numerical operation.

Definition
A logarithm is the power to which an arbitrary (but constant) number (the base) is raised to produce a given value.
EX: If "l" is the logarithm of the value "v" with base "b", then bl = v.

Application
Logarithms are commonly used to multiply the values which they represent by the process of adding the logarithms, which is simpler than multiplying the original values. In this process, we have :
a) the addition (procedure of stage 1), b) of several exponentials or logarithms (procedure of stage 3), c) to operate a multiplication of the original values (procedure of stage 2).
We therefore have the simpler procedure of stage 1 (addition), using the abbreviated language of stage 3 (exponentials), replacing the more complex procedure of stage 2 (multiplication). This equivalence of operations, involving the three stages, is the result of the isomorphism which exists between the series of stage 1 and stage 2, and the operation of stage 3 as an abbreviation of stage 2, as illustrations of the formula bl = v.
Series of stage 1, the logarithm "l" (increment 1): ... -3 -2 -1 0 1 2 3 ...
Series of stage 2, the value "v" (increment 2): ... 1/8 1/4 1/2 1 2 4 8 ...
Abbreviation of the elements of stage 2 using the language of stage 3, with base "b" (2) : ... 2-3 2-2 2-1 20 21 22 23 ...

 Series of stage 1, the logarithm ... -3 -2 -l 0 1 2 3 ... Series of stage 2, the value ... 1/8 1/4 1/2 1 2 4 8 ... Abbreviation stage 2 using stage 3 ... 2-3 2-2 2-l 20 2l 22 23 ...

Isomorphisms
If we line up the series of stage 1 and stage 2 so that their central points coincide (the value 0 of stage 1 directly above the value 1 of stage 2), we notice that the values of the first series (stage 1) always correpond exactly to the power in the third series (abbreviation of the values of stage 2 when they are presented in the language of stage 3).
EX: the value 2 of stage 2 is 21 (two, once) - see the number 1 in stage 1 ;
the value 4 of stage 2 is 22 (two, twice) - see the number 2 in stage 1 ;
the value 8 of stage 2 is 23 (two, three times) - see the number 3 in stage 1 ; and so on ...

Verification
All the isomorphisms of the right hand section of these series (the part with whole and positive numbers) are easily verified in this manner. On the other hand, there does not seem to be any similar verification for the central position or for the values on the left of the series (the values in fractions and negative numbers).
We see from the isomorphism of the series that :
a) 2 to the power 0 equals 1 ;
b) 2 to the power -1 equals 1/2 ;
c) 2 to the power -2 equals 1/4 ;
d) 2 to the power -3 equals 1/8 ;
and so on...

These relationships exist
But we have no verification for these answers, at least not the evident, tangible verification that we had for the values on the right hand side of the series.
How can one imagine a value raised to the power 0? or to a negative power?
However, we cannot deny that these relationships exist as part of our most elementary mathematics.

Working back
It is possible to deduce the values of zero and negative powers by using the inverse procedure, that of lowering the powers gradually :
1) Each time that we added a unit to the power, we multiplied the base (or the preceding value) by the increment (in this case, by 2), EX :
21 is 2 once (= 2)
22 is 21 x 2 (= 4)
23 is 22 x 2 (= 8)
2) By reversing this process, each time that we subtract a unit from the power, we shall divide the preceding number by the increment (always 2), EX :
23 = 8
22 is 23divided by 2 (= 4)
21 is 22 divided by 2 (= 2)
20 is 21 divided by 2 (= 1)
2-1 is 20 divided by 2 (= 1/2)
2-2 is 2-1 divided by 2 (= 1/4)
and so on ...

One must also admit deduced truths
This understanding of zero and negative exponents will enable us to present two principles which seem of the greatest importance and most relevant to the work which will follow :
1. It is not advantageous to limit oneself to "verifiable" truths; one must also admit "deduced truths" if they are the outcome of perfectly regular systems of which at least a part has been verified.
2. On the other hand, one must be wary of "deduced truths" if one cannot justify the system from which they spring or if one cannot at least offer verification for part of the operation of that system.
In closing, let us note that it seems preferable, since the operation of logarithms takes place at three levels of stages, not to use the term "logarithmic" to denote a particular stage (or its series) ; this term is frequently used to describe stage 2 but it seems far preferable to use the term "multiplicative" or even the term "geometric".