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From: k.hagan_at_[hidden]
Date: 2001-07-02 04:43:09

There still seems to be some discussion to the effect that decibels
and pH values or the arguments to exp() and log() might have units,
so I'd like to make two points.

Firstly, I would put money on the fact that any physicist who saw a
units library that handled decibels as a unit would smile and surf
over to Fermilab to use a library written by someone who "knew what
they were doing". For anyone trained as a physicist, it's a no-
brainer. (Looking to any future standardisation, you'll never get
decibels past all the national bodies.)

Secondly, someone has already posted to explain why. One can expand
exp(x) as a Taylor series in powers of x. Therefore, exp(x) has units
of x + x^2 + x^3 + ... The only "x" that fits the bill is
dimensionless. Consequently, log(x) must also be dimensionless
because log(exp(x))==x. Similary arguments apply to all
transcendental functions.

If economists habitually use formulae where exp(x) sits on the right
and some dimensioned value sits on the left then that's fine. The
true formula must be something like...

   LHS = c * RHS

...where c is numerically equal to 1 but carries the units of the
left hand side. In such circumstances, I might not write "c" either,
but I would strongly maintain that it is there. If you disagree, just
measure the LHS in different units, and c no longer ==1.

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