LOGARITHM is one of the two distinct INVERSES of EXPONENTIATION: log_ab = c iff b^c = a. But log_ba is IRRATIONAL ("a NONREPEATING sequence or decimal") WHENEVER numbers a, b are COPRIME (have no shared factor). This renders LOGARITHM (EXPONENTIAL INVERSE) PARTIAL over RATIONALS. We need it TOTAL.

By adjoining, to the FINITARY OPERATIONS (addition, multiplication, etc.) OF ARITHMETIC, the TRANSFINITARY OPERATION of LIMIT, we can DEFINE CAUCY SEQUENCES to REPRESENT ALL LOGARITHMS, RENDERING THIS OPERATION TOTAL.

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DEFINITION: A metric is a nonnegative function g(x,y) describing the "Distance'' (difference between coordinate values) between neighboring points for a given Set. A metric satisfies the Triangle Inequality, g(x,y) + g(y,z) ³ g(x,z), and is symmetric: g(x,y) = g(y,x).

DEFINITION: A Cauchy sequence, {a₁, a₂, a₃, ..., is a sequence such that its metric, d(a_m,a_n) satisfies
lim_{min(m,n)→¥} d(a_m,a_n).

Strictly speaking, these are INFINITE VECTORS OF RATIONALS. But the DECIMAL REPRESENTATION OF RATIONALS ENABLES US TO REDUCE ALL SUCH VECTORS TO THREE TYPES:

1. log_ba = {n, n, n, n, ...}, which is obviously INTEGRAL.
2. log_ba < 1, a "pseudo-EGYPTIAN" type, which can be REPRESENTED or APPROXIMATED by a DECIMAL NUMBER with CHARACTERISTIC ZERO.
3. Exceptions to the first two types.

But, clearly, THE DECIMAL REPRESENTATION encompasses all THREE TYPES OF INFINITE VECTORS, HENCE HIDES THE VECTOR NATURE.

However, again, we haven't abandoned the NATURAL NUMBERS from which we took our start. EVERY REAL NUMBER IS EQUIVALENT TO

• AN INFINITE VECTOR OF THE FORM, [r₁, r₂, r₃, ....], WHERE THE r_i ARE RATIONALS, AND THE SEQUENCE IS A CAUCHY SEQUENCE (WITH LIMIT);
• EACH RATIONAL, r_i, IS EQUIVALENT TO AN ORDERED PAIR OF ORDERED PAIRS, [[a,b], [c,d], WHERE a,b,c,d ARE NATURAL NUMBERS. REALS ARE SIMPLY NATURALS ON A TRANSFINITE LEVEL.

SO WE HAVE ONE KIND OF INVERSE (LOGARITHMIC) FOR EXPONENTIATION.

Now, let's see the other INVERSE for EXPONENTIATION.