Use Ito's Lemma to compute $d(\log S(t))$ and use this to find the closed form solution of S(t) I am having issues with this practice problem. If someone could help me solve it that would be greatly appreciated!

Let $S(t)$ be the stock price that satisfies the BSM model in SDE form
  $$dS(t) = \mu S(t) dt + \sigma S(t) dW_t$$ 
  where $\mu > 0$ and $\sigma > 0$ are two constants. Use Ito's Lemma to compute $d \log S(t)$ and use this result to find the closed form solution of $S(t)$.

 A: As the title of the question says, it's just a straightforward application of Ito's lemma:
Since $S$ satisfies the given SDE, $(\log)'(x)=x^{-1}$, and $(\log)''(x)=-x^{-2}$, we have
$$
\begin{split}
d(\log (S_t))&= \frac{1}{S(t)}dS_t-\frac{1}{2}\frac{1}{S_t^2}\sigma^2S_t^2dt\\
&= \mu dt+\sigma dW_t-\frac{\sigma^2}{2}dt\\
&=(\mu-\frac{\sigma^2}{2})dt+\sigma dW_t
\end{split}
$$
And this is just the SDE of the Brownian with drift.
A: Let $S(t)$ be governed by the SDE 
$$dS(t)=\mu S(t)dt+\sigma S(t)dW_t$$
Let $f(S)=\log(S)$.
Heuristically, we can write
$$\begin{align}
d\log(S)&=\frac{\partial f(S)}{\partial t}\,(dt)+\frac{\partial f(S)}{\partial S}\,(dS)+\frac12\frac{\partial ^2f(S)}{\partial S^2}(dS)^2\\\\
&=\frac{\partial \log(S)}{\partial t}\,(dt)+\frac{\partial \log(S)}{\partial S}\,(dS)+\frac12\frac{\partial ^2\log(S)}{\partial S^2}(dS)^2\\\\
&=(0)\,dt+\frac1S\,(dS)-\frac{1}{2S^2}(dS)^2\\\\
&=\left(\mu-\frac12 \sigma^2\right)\,dt+\sigma dW_t \tag 1
\end{align}$$
Now, integrating both sides of $(1)$ yields
$$\begin{align}
\int_0^t d\log(S(t'))&=\log(S(t)/S(0)\\\\
&=\int_0^t\left(\left(\mu-\frac12 \sigma^2\right)\,dt'+\sigma dW_t'\right)\\\\
&=\left(\mu-\frac12 \sigma^2\right)t+\sigma (W(t)-W(0))\\\\
S(t)=&S(0)e^{\left(\mu-\frac12 \sigma^2\right)t+\sigma (W(t)-W(0))}
\end{align}$$
