Expert: Paul Klarreich - 11/10/2011

Question
Hello Sir,
I believe this set of functions ( a^(1x),a^(2x),....,a^(nx))
are linearly independent.Given it is equivalent to
( (a^(x))^1,(a^(x))^2,....,(a^(x))^n);
and if var=a^x;//where a is any is integer;
( (var)^1,(var)^2,....,(var)^n);
1)Correct?
2)Is it not true that any two exponential functions can only share one point?

Answer
Questioner: sean
Country: Gambia
Category: Calculus
Private: No
Subject: linear independence of exponential functions
Question: Hello Sir,
I believe this set of functions ( a^(1x),a^(2x),....,a^(nx))
are linearly independent.Given it is equivalent to
( (a^(x))^1,(a^(x))^2,....,(a^(x))^n);
and if var=a^x;//where a is any is integer;
( (var)^1,(var)^2,....,(var)^n);
1)Correct?

Yes, this looks OK.

2)Is it not true that any two exponential functions can only share one point?

If you mean OF THIS TYPE, then I agree.
Take two functions, which, WLOG (as we say), can be:

exp(Nx)  and  exp(Mx)

Set them equal and try to solve for x.  Show that if M /= N,  there is only one solution.