Editing Surreal number (section)

===Results===
Using this definition, the following hold:{{efn|Even the most trivial-looking of these equalities may involve transfinite induction and constitute a separate theorem.}}
* {{math|exp}} is a strictly increasing positive function, {{math|1=''x'' < ''y'' ⇒ 0 < exp ''x'' < exp ''y''}}
* {{math|exp}} satisfies {{math|1=exp(''x'' + ''y'') = exp ''x'' · exp ''y''}}
* {{math|exp}} is a [[surjection]] (onto <math display=inline>\mathbb{No}_+</math>) and has a well-defined inverse, {{math|1=log = exp{{sup|–1}}}}
* {{math|exp}} coincides with the usual exponential function on the reals (and thus {{math|1=exp 0 = 1, exp 1 = ''e''}})
* For {{mvar|x}} infinitesimal, the value of the formal power series ([[Taylor expansion]]) of {{math|exp}} is well defined and coincides with the inductive definition
** When {{mvar|x}} is given in Conway normal form, the set of exponents in the result is well-ordered and the coefficients are finite sums, directly giving the normal form of the result (which has a leading {{math|1}})
** Similarly, for {{mvar|x}} infinitesimally close to {{math|1}}, {{math|log ''x''}} is given by power series expansion of {{math|''x'' – 1}}
* For positive infinite {{mvar|x}}, {{math|exp ''x''}} is infinite as well
** If {{mvar|x}} has the form {{mvar|ω{{sup|α}}}} ({{math|''α'' > 0}}), {{math|exp ''x''}} has the form {{mvar|ω{{sup|ω{{sup|β}}}}}} where {{mvar|β}} is a strictly increasing function of {{mvar|α}}. In fact there is an inductively defined bijection <math display=inline>g: \mathbb{No}_+ \to \mathbb{No} : \alpha \mapsto \beta</math> whose inverse can also be defined inductively
** If {{mvar|x}} is "pure infinite" with normal form {{math|1=''x'' = Σ{{sub|''α''<''β''}}''r''{{sub|''α''}}''ω''{{sup|''a''{{sub|''α''}}}}}} where all {{math|1=''a''{{sub|''α''}} > 0}}, then {{math|1=exp ''x'' = ''ω''{{sup|Σ{{sub|''α''<''β''}}''r''{{sub|''α''}}''ω''{{sup|''g''(''a''{{sub|''α''}})}}}}}}
** Similarly, for {{math|1=''x'' = ''ω''{{sup|Σ{{sub|''α''<''β''}}''r''{{sub|''α''}}''ω''{{sup|''b''{{sub|''α''}}}}}}}}, the inverse is given by {{math|1=log ''x'' = Σ{{sub|''α''<''β''}}''r''{{sub|''α''}}''ω''{{sup|''g''{{sup|–1}}(''b''{{sub|''α''}})}}}}
* Any surreal number can be written as the sum of a pure infinite, a real and an infinitesimal part, and the exponential is the product of the partial results given above
** The normal form can be written out by multiplying the infinite part (a single power of {{mvar|ω}}) and the real exponential into the power series resulting from the infinitesimal
** Conversely, dividing out the leading term of the normal form will bring any surreal number into the form {{math|1=(''ω''{{sup|Σ{{sub|''γ''<''δ''}}''t''{{sub|''γ''}}''ω''{{sup|''b''{{sub|''γ''}}}}}})·''r''·(1 + Σ{{sub|''α''<''β''}}''s''{{sub|''α''}}''ω''{{sup|''a''{{sub|''α''}}}})}}, for {{math|1=''a''{{sub|''α''}} < 0}}, where each factor has a form for which a way of calculating the logarithm has been given above; the sum is then the general logarithm
*** While there is no general inductive definition of {{math|log}} (unlike for {{math|exp}}), the partial results are given in terms of such definitions. In this way, the logarithm can be calculated explicitly, without reference to the fact that it's the inverse of the exponential.
* The exponential function is much greater than any finite power
** For any positive infinite {{mvar|x}} and any finite {{mvar|n}}, {{math|1=exp(''x'')/''x''{{sup|''n''}}}} is infinite
** For any integer {{mvar|n}} and surreal {{math|''x'' > ''n''{{sup|2}}}}, {{math|exp(''x'') > ''x''{{sup|''n''}}}}. This stronger constraint is one of the Ressayre axioms for the real [[exponential field]]<ref name=vdDE2001 />
* {{math|exp}} satisfies all the Ressayre axioms for the real exponential field<ref name=vdDE2001 />
** The surreals with exponential is an [[elementary extension]] of the real exponential field
** For {{math|''ε''{{sub|''β''}}}} an ordinal epsilon number, the set of surreal numbers with birthday less than {{math|''ε''{{sub|''β''}}}} constitute a field that is closed under exponentials, and is likewise an elementary extension of the real exponential field