f(x+y)= f(x)f(y)

If we take log for both sides,

we have g(x)= g(x) + g(y) by letting g=log f

Then, after a series of calculation and checking,

we have f(x)= a^ (cx)

I have a few questions dealing with the log.

Can I choose any base, for example 10?

Must a=e where it is the natural log.?

Last, I am confused with the use of fixed point, or perhaps I even don't know the complete definiton of me, hence, finding myself incapable of doing this kind of calculations.

EDIT: I am very interested and enthusiastic for functional equation.

However, I can't find any books related to this topic or subtopic.

I've checked algebra, statistics. Does it belong to Discrete maths or analysis or other areas?

When in doubt, Mathworld http://mathworld.wolfram.com/FunctionalEquation.html

 

Note in the References three books by Kuczma on functional equations.

When in doubt just google. That link is the first hit.

f(x+y)= f(x)f(y)

If we take log for both sides' date='

we have g(x)= g(x) + g(y) by letting g=log f[/quote']

 

 

do we? What happens if you subtract g(x) from both sides?

 

 

Then, after a series of calculation and checking,

we have f(x)= a^ (cx)

I have a few questions dealing with the log.

Can I choose any base, for example 10?

Must a=e where it is the natural log.?

Last, I am confused with the use of fixed point,

 

what fixed point? you haven't mentioned a fixed point.

 

or perhaps I even don't know the complete definiton of me...

 

which of us truly knows the definition of themselves?

We should point out that you've omitted to mention at least one important property that you're assuming f must have, namely continuity.

 

It is trivial to show that log(f(x)) for rational x is completely determined by f(1) and hence for rational x, f(x)=exp{kx} for some constant k, and thus by continuity f(x) is exponentiation for all x *if* we assume f is continous. If f is not continuous then there are uncountably many distinct f's with this multiplicative property.

we have g(x)= g(x) + g(y) by letting g=log f

sorry it is g(x+y) for the left-hand side

It is trivial to show that log(f(x)) for rational x is completely determined by f(1) and hence for rational x, f(x)=exp{kx} for some constant k, and thus by continuity f(x) is exponentiation for all x *if* we assume f is continous. If f is not continuous then there are uncountably many distinct f's with this multiplicative property.

Oh I now realize the importance of the continuity.

Must we take e as the base of log?

Where does anything require you to use any particular base? logs in different bases differ only by constants. This is why at the end you only write exp{kx{ for some constant k. k can be gotten from f(1).

OK thanks