(1st Post) Estimating the transformation of a gamma random variable with a lognormal distribution

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If X is gamma(n,$\lambda$) distributed, what should $\alpha$ and $\beta$ be such that the constructed random variable $\beta*exp(\alpha X)$ is approximated by the lognormal(0,1) distribution when n is sufficiently large. I have tried splitting the gamma variable into a sum of exponential variables and also tried Taylor expanding $\beta*exp(\alpha X)$ but neither seems to be working.