A dataset contains $n$ values. Let $\mu$ be the mean and $\sigma^2$ be the variance. If we define a new set of values $y_i = \ln(x_i)$ and assume the original values $x_i$ are very close to the mean $\mu$, use the Taylor expansion of $\ln(x)$ around $\mu$ to estimate the variance of the log-transformed data, $\text{Var}(Y)$.