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 between addition and multiplication JmsNxn Long Time Fellow Posts: 291 Threads: 67 Joined: Dec 2010 03/16/2011, 12:20 AM (This post was last modified: 03/16/2011, 12:28 AM by JmsNxn.) If the operators over {q}, 0<=q<=1, are defined with an identity given by S(q), then they can define tetration if we allow: q:log(x) = exp^[-q](x) (1) x {q} y = -q:log(q:log(x) + q:log(y)) (2) x {1+q} y = -q:log(q:log(x) * y) you yourself discussed these operators, I was pleasantly surprised to see someone come to the same formula as me. Therefore, when I was talking about solving for tetration I was talking about defining x {q} y numerically as this modified Gauss mean and then conversely also allowing the two laws of logarithmic semi-operators (1). This would imply evaluations of rational values for tetration since semi-operators depend on tetration given (1) and (2). Sadly however, this pseudo Gauss mean yields no identity, or no universal value S(q) for all x E C such that: x {q} S(q) = x this reduces the logarithmic laws (1) & (2) as null since q:log(S(q)) = 0 is an essential identity, and there is no value S(q) (03/16/2011, 12:18 AM)tommy1729 Wrote: perhaps x {y} z is the mean ONLY for 0 < y < 1 and x {y} z for y > 1 is simply the (y-floor(y)) th iteration of x {floor(y)} z. by lloyd rational exponentiation, or {y} for 1