# Exponent Law proof Cant find anywhere?

## Main Question or Discussion Point

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

Last edited:

lurflurf
Homework Helper
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

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

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
Homework Helper
The best formal proof would be by induction.

lurflurf
Homework Helper
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