Show that $f = 0$ if $\int_a^b f(x)e^{kx}dx=0$ for all $k$ The problem is show that $f=0$ whenever $f\in C[a,b]$ and
$$\int_a^bf(x)e^{kx}dx =0, \hspace{1cm}\forall k\in\mathbb{N}.$$
Can someone help me?
Thank you!

First, letting $u=e^x$ we note that $\int_{e^a}^{e^b}f(\ln u)u^{k-1}du=0$ for all $k\in\mathbb N$.
Next, see the following old questions:
Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
If $f$ is continuous on $[a , b]$ and $\int_a^b f(x) p(x)dx = 0$ then $f = 0$
problem on definite integral
