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If
then

And this can be proved using Ito's lemma, but the following gives a different solution.
From:

divide throughout by S_t:
dS_t/S_t = mu dt + sigma dW_t

divide throughout by dt:

dS_t/dt * 1/ S_t = mu + sigma dW_t/dt

integrate both sides with respect to dt and limits 0 and t:

ln(S_t) - ln(S_0) = mu*t + sigma dW_t

add ln(S_0) to both sides:

ln(S_t) = mu*t + sigma dW_t + ln(S_0)

take the exponential of both sides:

S_t = exp(mu*t + sigma dW_t + ln(S_0))

S_t = exp(mu*t + sigma dW_t) * exp(ln(S_0)))

S_t = S_0 * exp(mu*t + sigma dW_t)

which is not the same solution that Ito's Lemma gives, what am I doing incorrectly?

Any help much appreciated.