A simple question regarding the derivation of the Black-Scholes formula

Question

i am taking a derivatives class, and what is of course obligatory is to derive the Black Scholes formula. I am simply stuck or puzzled by one simple thing in the derivation:

How do they get from $$exp(y-\frac{(y-logS_0 - (r - 1/2 \sigma^2)T)^2}{2\sigma^2 T})$$

to

$$S_0e^{rT} exp(-\frac{(y-logS_0 - (r + 1/2 \sigma^2)T)^2}{2\sigma^2 T})$$

Answer 1

For the equation to be true, using the property that $\exp(a+b)=\exp(a)\exp(b)$,

it is true if

$$\exp(y)=S_o\exp(rT)$$

That is if $y$ is defined to be the logarithm of the value after time $T$ with interest rate $r$.

That is there must be a relationship between $y$ and $S, r,T$.