text: Bridgman’s theorem restricts the type of function that can be used to define a physical quantity from general (dimensionally compounded) quantities to only products of powers of the quantities, unless some of the independent quantities are algebraically combined to yield dimensionless groups, whose functions are grouped together in the dimensionless numeric multiplying factor.  This excludes polynomials of more than one term or transcendental functions not of that form.
  Scalar arguments to transcendental functions such as exponential, trigonometric and logarithmic functions, or to inhomogeneous polynomials, must be dimensionless quantities.  (Note: this requirement is somewhat relaxed in Siano's orientational analysis described below, in which the square of certain dimensioned quantities are dimensionless.)
  While most mathematical identities about dimensionless numbers translate in a straightforward manner to dimensional quantities, care must be taken with logarithms of ratios: the identity , where the logarithm is taken in any base, holds for dimensionless numbers  and , but it does not hold if  and  are dimensional, because in this case the left-hand side is well-defined but the right-hand side is not.
  Similarly, while one can evaluate monomials () of dimensional quantities, one cannot evaluate polynomials of mixed degree with dimensionless coefficients on dimensional quantities: for , the expression  makes sense (as an area), while for , the expression  does not make sense.
  However, polynomials of mixed degree can make sense if the coefficients are suitably chosen physical quantities that are not dimensionless.  For example,
  This is the height to which an object rises in time  if the acceleration of gravity is 9.8  and the initial upward speed is 500 .  It is not necessary for  to be in seconds.  For example, suppose  = 0.01 minutes.  Then the first term would be