I am trying to calculate the expectation $$E[e^{cX}]$$ for arbitrary $c<0$ (for $c>0$ the expectation is infinite) if $X$ is lognormally distributed, i.e. $\log(X) \sim N(\mu, \sigma)$.
My idea was to write the expectation as an integral, but I did not see how to proceed: $$E[e^{cX}] = \frac{1}{\sqrt{2\sigma\pi}}\int_0^\infty \frac{1}{x}\exp\left(cx - \frac{(\log x - \mu)^2}{2\sigma^2}\right)dx$$
I also tried the Itô formula (the actual task is to find $E[e^{cX_T} \mid X_t = x]$ where $X$ is a geometric Brownian motion, but it reduces to the problem above because we are looking at a Markov process), but that did not look very promising either. Can anybody help me?