Log-price dynamics under geometric Brownian motion

A stock follows $dS_t=\mu S_t\,dt+\sigma S_t\,dW_t$. Use Ito's formula to find the stochastic differential of $\log S_t$.

$d\log S_t=\big(\mu-\tfrac12\sigma^2\big)dt+\sigma\,dW_t$. With $f=\log x$, $f'=1/x$ and $f''=-1/x^2$, and $(dS)^2=\sigma^2S^2dt$, so the second-order term subtracts $\tfrac12\sigma^2\,dt$. The log-price is therefore a Brownian motion with drift, which is why $S_T$ is lognormal.