SAT Question 263f9937: Nonlinear Functions answer and explanation with Desmos solution

Choice D is correct. The number of bacteria per milliliter is doubling each day. For example, from day $1$ to day $2$, the number of bacteria increased from $2.5\times 10^5$ to $5.0\times 10^5$. At the end of day $3$ there are $10^6$ bacteria per milliliter. At the end of day $4$, there will be $10^6\times 2$ bacteria per milliliter. At the end of day $5$, there will be $(10^6\times 2)\times 2$, or $10^6\times 2^2$ bacteria per milliliter, and so on. At the end of day $d$, the number of bacteria will be $10^6\times 2^{d-3}$. If the number of bacteria per milliliter will reach $5.12\times 10^8$ at the end of day $d$, then the equation $10^6\times 2^{d-3}=5.12\times 10^8$ must hold. Since $5.12\times 10^8$ can be rewritten as $512\times 10^6$, the equation is equivalent to $2^{d-3}=512$. Rewriting $512$ as $2^9$ gives $d-3=9$, so $d=12$. The number of bacteria per milliliter would reach $5.12\times 10^8$ at the end of day $12$.