If I wanted something to decay with a half-life of $n$ days, what should I multiply it by every day? I feel like this should have a rather simple solution using logarithms, but I don't quite know where to start.
Basically, every day I am going to multiply a given quantity by $r$, where $r < 1$.
Obviously, if $r$ is 0.5, then my half-life $n$ is 1 day. (This is also stated to remove any ambiguities on what counts as a day/range)
What is the relationship between $r$ and $n$?
 A: Suppose $x$ is the quantity, and you want it to decay to $\frac{1}{2}$ after $n$ discrete steps (whether they be minutes, hours, days, years, eons, or nanoseconds), and you want to model that by multiplication by $r$.
You start with $x$. After one time interval, you want to have $rx$. After two time intervals, you want $r(rx) = r^2x$. After three time intervals, $r(r^2x)=r^3x$. After $n$ times intervals, $r^nx$. Since you want $r^nx = \frac{1}{2}x$, you can now solve for $r$.
This is discrete decay, though, not continuous decay. It is exactly analogous to the difference between continuous interest compounding, and interest compounding in discrete time intervals. "Half-life" is a term usually reserved for continuous decay, so you'll want to be careful with that.
A: Suppose you start with a quantity of 1 (whatever that means).  You know that after $n$ days, there should be $\frac{1}{2}$ left.  This should give you an equation that you can solve for $r$ (in terms of $n$ or with a specific value of $n$).
