  Inspired by Rearick (1968), we introduce two new operators, LOG and EXP. The
LOG operates on generalized Fibonacci polynomials giving generalized Lucas
polynomials. The EXP is the inverse of LOG. In particular, LOG takes a
convolution product of generalized Fibonacci polynomials to a sum of
generalized Lucas polynomials and EXP takes the sum to the convolution product.
We use this structure to produce a theory of logarithms and exponentials within
arithmetic functions giving another proof of the fact that the group of
multiplicative functions under convolution product is isomorphic to the group
of additive functions under addition. The hyperbolic trigonometric functions
are constructed from the EXP operator, again, in the usual way. The usual
hyperbolic trigonometric identities hold. We exhibit new structure and
identities in the isobaric ring. Given a monic polynomial, its infinite
companion matrix can be embedded in the group of weighted isobaric polynomials.
The derivative of the monic polynomial and its companion matrix give us the
different matrix and the infinite different matrix. The determinant of the
different matrix is the discriminate of the monic polynomial up to sign. In
fact, the LOG operating on the infinite companion matrix is the infinite
different matrix. We prove that an arithmetic function is locally representable
if an only if it is a multiplicative function. An arithmetic function is both
locally and globally representable if it is trivially globally represented.
