Question 135981
1. A half life equation has the form of 
{{{N(t) = N[0]e^(-(alpha)t)}}}
and you calculate {{{N[0]}}} and {{{alpha}}} using the clues provided. 
At time t=0, the value of N(0)=100. 
{{{N(0)=N[0]e^(0)=100}}}
{{{N[0]=100}}}
since {{{e^(0)=1}}}.
Also, at t=5730, N(t)=50. 
{{{50=100e^(-(alpha)5730)}}}
{{{0.5=e^(-(alpha)5730)}}}
{{{ln(0.5)=-(alpha)5730)}}}
{{{-0.693147=-(alpha)5730)}}}
{{{(alpha)=0.693147/5730)}}}

{{{(alpha)=0.00012097)}}}
{{{N(t) = 100e^(-(.00012097)t)}}}
2. Now that you have the equation, calculations are easy.
{{{N(t) = 100e^(-(.00012097)t)}}}
{{{48.37 = 100e^(-(.00012097)t)}}}
{{{.4837 = e^(-(.00012097)t)}}}
{{{ln(.4837) = -(.00012097)t)}}}
{{{.7262904=.00012097t}}}
{{{t=6003.888}}}