Question 999529
The first function formula you show is a good choice, intended for continuous decay.  Take logarithms of both sides and solve for k, and reuse some steps to find half-life.


For simpler work, use R to mean R(t).
{{{R(t) = re^(-kt)}}}
{{{ln(R)=ln(r)-kt}}}
{{{ln(R)-ln(r)=-kt}}}
{{{kt=ln(r)-ln(R)}}}
{{{highlight_green(k=ln(r/R)/t)}}}


The description gave data for the one-tenth life, and you want to know half-life.  Use the formula for k.


{{{k=ln(10/1)/30}}}
{{{highlight_green(k=69)}}}.



FIND HALF-LIFE


{{{R=re^(-kt)}}}, basic decay equation 
{{{R=re^(-69t)}}}
but you have a step already which gives the formula you want to find half-life.


{{{kt=ln(r/R)}}}----from earlier steps
{{{highlight_green(t=ln(r/R)/k)}}}
Values to use  {{{system(r=1,R=1/2,k=69)}}}
Now evaluate t for half-life.