Question

Prove: a^b*a^c=a^(b+c) for real a, b, c.

Answer 1

\[a,b,c \in \mathbb{R}\]\[Prove: a^{b}a^{c}=a^{b+c}\]

Answer 2

Wait ! a should be a positive real !

Answer 3

a>1
I forgot to say that. Facepalm.

Answer 4

It is true for all a>0.
We can say :
\[\Large a^ba^c=e^{\ln a^b}e^{\ln a^c}\\
~~~~~~~~~\Large =e^{b\ln a}e^{c\ln a}\\
~~~~~~~~~\Large=e^{b\ln a+c\ln a}\\
~~~~~~~~~~\Large=e^{(b+c)\ln a}\\
~~~~~~~~~~\Large=e^{\ln a^{b+c}} \\
~~~~~~~~~~\Large=a^{b+c}\]

Answer 5

I can't use logs. I'm doing analysis and I'm still proving the fundamentals of real powers from the field axioms.

Answer 6

OK ! If b and c are natural integers you can prove it using induction !

Answer 7

b, c are just real.

Answer 8

@mitodoteira My last reply is the 1st step of the proof !

Answer 9

How?

Answer 10

I dont think we can use induction here... three variables

Answer 11

Isnt 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.

Answer 12

No, it isn't a property.

Answer 13

hmmmmm how do you define the exponential function?

Answer 14

http://math.stackexchange.com/questions/435751/proving-the-product-rule-for-exponents-with-the-same-base
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.