Exponent Law proof Cant find anywhere?

[tamintl]

I have been studying the exponent laws in depth and I cannot find a formal proof on this law:

exp(a+b) = exp(a)exp(b)

Is it formal enough to say the following? (I think not since I give an example with numbers in it)

We define:
a² = a*a.
For example:
a^8 a^7 = a*a*a*a*a*a*a*a a*a*a*a*a*a*a = a^15 = a^8+7

More generally:
a^(c+b) = a^c * a^b = a^(b+c)

Regards as always
Tam

 

[lurflurf]

Proofs of this type depend upon your definition of exp.

I like to take the addition formula as the definition, so the proof is obvious from the definition.

Others define exp with a limit, a series, an inverse of an integral, or a differential equation and prove it by manipulating whatever their definition is.

Your proof can be extended to prove it for all rationals. The extension to reals is done by taking limits

 

[tamintl]

Proofs of this type depend upon your definition of exp.

I like to take the addition formula as the definition, so the proof is obvious from the definition.

Others define exp with a limit, a series, an inverse of an integral, or a differential equation and prove it by manipulating whatever their definition is.

Your proof can be extended to prove it for all rationals. The extension to reals is done by taking limits

Thanks for your answer.

I'd like to take the proof from this definition: exp(x) = lim (n→∞) (1 + x/n)ⁿ (Limit)

Would you be able to point my in the right direction please?

Regards
Tam

 

[HallsofIvy]

The best formal proof would be by induction.

 

[lurflurf]

Try to show that
exp(a)exp(b)/exp(a+b)=1
starting with
exp(a)exp(b)/exp(a+b)=
lim (n→∞) (1 + a/n)ⁿ lim (n→∞) (1 + b/n)ⁿ /lim (n→∞) (1 + (a+b)/n)ⁿ =
lim (n→∞) [(1 + a/n)(1 + b/n)/(1 + (a+b)/n)]ⁿ
and so on