Editing Exponential function (section)

===Properties===
''[[multiplicative inverse|Reciprocal]]:'' The functional equation implies {{tmath|1=e^x e^{-x}=1}}. Therefore {{tmath|e^x \ne 0}} for every {{tmath|x}} and 
<math display=block>\frac 1{e^x}=e^{-x}.</math>

''Positiveness:'' {{tmath|e^x>0}} for every real number {{tmath|x}}. This results from the [[intermediate value theorem]], since {{tmath|1=e^0=1}} and, if one would have {{tmath|e^x<0}} for some {{tmath|x}}, there would be an {{tmath|y}} such that {{tmath|1=e^y=0}} between {{tmath|0}} and {{tmath|x}}. Since the exponential function equals its derivative, this implies that the exponential function is [[monotonically increasing]].

''Extension of [[exponentiation]] to positive real bases:'' Let {{mvar|b}} be a positive real number. The exponential function  and the natural logarithm being the inverse each of the other, one has <math>b=\exp(\ln b).</math> If {{mvar|n}} is an integer, the functional equation of the logarithm implies 
<math display=block>b^n=\exp(\ln b^n)= \exp(n\ln b).</math>
Since the right-most expression is defined if {{mvar|n}} is any real number, this allows defining {{tmath|b^x}} for every positive real number {{mvar|b}} and every real number {{mvar|x}}:
<math display=block>b^x=\exp(x\ln b).</math>
In particular, if {{mvar|b}} is the [[Euler's number]] <math>e=\exp(1),</math> one has <math>\ln e=1</math> (inverse function) and thus <math display=block>e^x=\exp(x).</math> This shows the equivalence of the two notations for the exponential function.