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→Extension to real numbers: When this is defined for 0<''x''<1 the whole function easily follows for all ''x''>-2 |
If, for example 10^^0.5 = log 2, then 10^^1.5 = 2, 10^^ 2.5 - 100, etc. |
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Note that when solving multiple-level exponentiation, the exponentiation is done at the deepest level first (in the notation, at the highest level). In other words:
:<math>\,\!2^{2^{2^2}} = 2^{(2^{(2^2)})} = 2^{(2^4)} = 2^{16} = 65,\!536</math>
:<math>2^{2^{2^2}}</math> is not equal to <math>\,\! \left({(2^2)}^2\right)^2 = 256</math>
There is no standard notation for tetration. The notations in which it can be written (some of which allow further iteration) include:
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*it is correct for natural numbers x
*a^^(b+1) =
*it is a smooth function, monotonically increasing
When this is defined for 0<''x''<1 the whole function easily follows for all ''x''>-2
If, for example 10^^0.5 = log 2, then 10^^1.5 = 2, 10^^ 2.5 - 100, etc.
See http://home.earthlink.net/~mrob/pub/math/ln-notes1.html#real-hyper4 for attempts to extend tetration to real numbers.
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