5. Logarithmic and Exponential Fields
Logarithmic Addition
Logarithmic
addition and subtraction (respectively, "accretion" and
"erosion") are logarithmic conjugates of plus and minus:
X ln[+] Y
= X # Y =
ln( e^{X} + e^{Y} ) =
"X along Y"
X ln[] Y
= X $ Y =
ln( e^{X}  e^{Y} ) =
"X beyond Y"
Logarithmic
multiplication is addition:
X
ln[*] Y = ln ( e^{X} * e^{Y} ) =
X+Y
Therefore
logarithmic addition is a field under +, with identity  ∞:
X
+ ( Y # Z ) = ( X + Y ) # ( X + Z )
X
+ ( Y $ Z ) = ( X + Y ) $ ( X + Z )

∞ #
X = X
X $ X =  ∞
Defining
logarithmic negation require using Euler's formula e^{πi} = 1.
Define
$X =
 ∞
$ X =
ln(e^{X}) = "beyond X"; then
$X =
X + $0
$0 = πi
"Beyond
zero equals pi times i." This formula refers, in five symbols, to the five
major branches of mathematics; analysis, arithmetic, logic, geometry, and
algebra.
We
can define log addition to other bases. Let
X #_{C } Y
= log_{C}( C^{X}
+ C^{Y} ) = ( (X lnC)#(Y lnC) )
/ lnC =
X [#/lnC] Y
X $_{C } Y
= log_{C}( C^{X}
 C^{Y} ) = ( (X lnC)$(Y lnC) )
/ lnC =
X [$/lnC] Y
If P_{1} = A b^{X} and P_{2}
= A b^{Y} then (P_{1}+P_{2}) = A b ^{Z} , where Z
= X #_{b} Y.
When
exponentials add, exponents logadd. If
b = e^{r}, then
Z = X
[#/r] Y = ( rX # rY ) / r =
X ln ^{r}\/+ Y
; logfermat addition!
In
the decibel scale, sound energy is proportional to 10^{D/10} , where D
is decibels; therefore if two sound energies add linearly, then the decibels
add logarithmically;
D_{12} = D_{1} #_{10^0.1} D_{2}
In
the Richter scale, seismic energy is proportional to 10^{R}. If
energies add, then;
R_{12} = R_{1} #_{10} R_{2}
In
the pH scale, concentration is proportional to
10^{  pH} ; therefore, if we mix equal volumes of two solutions
and no buffering occurs, so that the concentrations average, then;
pH_{12} = (
pH_{1} #_{0.1} pH_{2} ) + ln2
Exponential Multiplication
Exponential
multiplication (or "eruption") is * conjugated by exp:
X e^{*}
Y = X
@ Y = e ^{(
lnX * lnY )} = X^{ lnY } = Y^{
lnX}
Its
inverse operation is exponential division, or "subduction":
X
e^{¸} Y =
X ^{(1/lnY)}
Unlike
exponentiation, eruption is commutative and associative:
X
@ Y =
Y @ X
X
@ ( Y @ Z ) = ( X @ Y ) @ Z
*
is a field under @, with identities 1 and e:
X
@ ( Y * Z ) = ( X @ Y ) * ( X @ Z )
X
@ 1 = 1
X
@ e = X
X
@ e^{N} = X^{N}
X
@ Y^{N} = (X@Y)^{N}
We
can exponentiate eruption: X e^{@}
Y = e^{(lnX @ ln Y)}. This is a field
over @, with identities e and e^{e}. And so on; we get an
"Exponential Tower" of operations:
# ,
+ , *
, @ , e^{@}
, ...

each one of which is a field over the preceding operation, with identities
∞
, 0 , 1
, e
, e^{e} , ...
Conjecture: Continuous Exponential Tower: there exists a continuous,
invertible, analytic "superexponential" operator E on the positive
reals such that E(x+1) = e^{E(x)}.
Given such an E, we can define the continuous iterate of
exponentiation:
exp_{R}(X) = E(
E^{1}(X) + R )
So
exp_{1}(X) = lnX ; exp_{0}(X) =
X ; exp_{1}(X) = e^{X} ; exp_{2}(X) = e^{(e^X)}
; ...
Therefore
we can make the exponential tower continuous, thus:
X _{R}+
Y = exp_{R}( exp_{R}(X) + exp_{R}(Y)
)
= E (
E^{1}{ E[E^{1}(X)  R]
+ E[E^{1}(Y)  R] }+ R )
So _{1}+ _{ }= #
, _{0}+ =
+ , _{1}+ =
* , _{2}+ =
@ , ...
Each _{R}+ distributes over _{R1}+
, and distributes under _{R+1}+ .
Open
questions remain. For instance, how does
_{1/2}+ relate to +
and * ?
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