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mitodoteira
 one year ago
Best ResponseYou've already chosen the best response.0\[a,b,c \in \mathbb{R}\]\[Prove: a^{b}a^{c}=a^{b+c}\]

Noura11
 one year ago
Best ResponseYou've already chosen the best response.0Wait ! a should be a positive real !

mitodoteira
 one year ago
Best ResponseYou've already chosen the best response.0a>1 I forgot to say that. Facepalm.

Noura11
 one year ago
Best ResponseYou've already chosen the best response.0It is true for all a>0. We can say : \[\Large a^ba^c=e^{\ln a^b}e^{\ln a^c}\\ ~~~~~~~~~\Large =e^{b\ln a}e^{c\ln a}\\ ~~~~~~~~~\Large=e^{b\ln a+c\ln a}\\ ~~~~~~~~~~\Large=e^{(b+c)\ln a}\\ ~~~~~~~~~~\Large=e^{\ln a^{b+c}} \\ ~~~~~~~~~~\Large=a^{b+c}\]

mitodoteira
 one year ago
Best ResponseYou've already chosen the best response.0I can't use logs. I'm doing analysis and I'm still proving the fundamentals of real powers from the field axioms.

Noura11
 one year ago
Best ResponseYou've already chosen the best response.0OK ! If b and c are natural integers you can prove it using induction !

mitodoteira
 one year ago
Best ResponseYou've already chosen the best response.0b, c are just real.

Noura11
 one year ago
Best ResponseYou've already chosen the best response.0@mitodoteira My last reply is the 1st step of the proof !

sauravshakya
 one year ago
Best ResponseYou've already chosen the best response.0I dont think we can use induction here... three variables

sauravshakya
 one year ago
Best ResponseYou've already chosen the best response.0Isnt it the property??? I am not sure if it can be proven. example : we know a*b=b*a because its a property what how to Prove it.

mitodoteira
 one year ago
Best ResponseYou've already chosen the best response.0No, it isn't a property.

swissgirl
 one year ago
Best ResponseYou've already chosen the best response.0hmmmmm how do you define the exponential function?

swissgirl
 one year ago
Best ResponseYou've already chosen the best response.0http://math.stackexchange.com/questions/435751/provingtheproductruleforexponentswiththesamebase Here is the link to this particular question I asked on MSE. Take a look at both proofs. The proof using Least Upper Bound is more analytical and a touch harder to follow but I think that is the proof you are looking for.
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