Conjugates of Fields, 5 of 6; Logarithmic and Exponential Fields

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   P₁ = A b^X   and  P₂ = A b^Y  then   (P₁+P₂)  =  A b ^Z  , where Z  =  X #_b Y.

When exponentials add, exponents log-add. If  b = e^r,  then

Z  =   X [#/r] Y  =  ( rX # rY ) / r   =   X   ln ^r\/+  Y  ;  log-fermat 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;

D₁₂   =   D₁   #_10^0.1  D₂

In the Richter scale, seismic energy is proportional to 10^R. If energies add, then;

R₁₂   =   R₁   #₁₀  R₂

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;

pH₁₂   =   ( pH₁   #_0.1  pH₂ )  +  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 "super-exponential" 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  ;   exp₀(X)  =  X  ;  exp₁(X)   =  e^X  ;  exp₂(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 _R-1+ , and distributes under _R+1+ .

Open questions remain. For instance, how does  _1/2+  relate to  +  and  * ?