Advertisers generally assume that the rate at which people hear about a product is proportional to the number of people who have not yet heard about it. Suppose that the size of a community is $15,000$, that to begin with no one has heard about a product, but that after 6 days $1500$ people know about it. How long will it take for $2700$ people to have heard of it?
I came up with formula $$y=-1+e^{1.219t}$$ and this is right formula from answer $$y=15000 \left(1-e^{-0.018t} \right)$$ both of them increase in proportional manner and reach about $1500 $ after $6$ days. But when it comes to calculating time for $2700$ people, the answer is different.
I drew both graphs on Desmos and this is what I got:
So I just assume my formula doesn't actually has any converging limit which is, in this case $15000$. However, I am curious if my formula can be still valid even if there is no such limit point?
As an answer to the given question, your formula is not valid because it does not correctly represent the property "the rate at which people hear about a product is proportional to the number of people who have not yet heard about it".
Looking at your formula, we have $y = -1 + e^{1.219t}$, and the rate of change is given by $\dot{y} = 1.219 e^{1.219t} = 1.219(y + 1)$. If we try to interpret that in the context of the problem, your formula satisfies the property "the rate at which people hear about a product is proportional to one more than the number of people who have already heard about it".
This is a valid alternative model, although it has issues - as you noted, it will grow without bound even though we only have a finite community of people to reach.
To get the answer given, you would start by noting that if $y$ people in the community have heard about the product, that means that $15000-y$ people haven't heard about it. So "the rate at which people hear about a product" needs to be proportional to this value, i.e. $\dot{y} \propto 15000 - y$, or in other words $\dot{y} = k(15000 - y)$ for some constant $k$. We can then integrate that and use the two given points to get the formula for $y$ given in the answer.