Spread of a lognormal raised to the fourth

HardRandom Variables Statistics~7m

Suppose $\ln X \sim \mathcal N(0, 1)$ . Find $\operatorname{Var}(X^4)$ . The answer has the form $e^{a} - e^{b}$ for positive integers $a, b$ ; find $a + b$ .

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$\ln(X^4)=4\ln X\sim\mathcal N(0,16)$ , so $X^4$ is lognormal with $\sigma^2=16$ .
Lognormal variance $=(e^{\sigma^2}-1)e^{2\mu+\sigma^2}=e^{32}-e^{16}$ .

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Spread of a lognormal raised to the fourth

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