Question 1130481
Model is {{{P(t)=P[o]e^(-kt)}}}, and half life is value of t when {{{P(t)=(1/2)(P[o])}}}.


FIND k, first.
Some variables changes just for notational and keyboard conveniences.

y=p*e^(-kt)  instead of {{{P(t)}}} and {{{P[o]}}};

{{{y=1*e^(-kt)}}}

{{{1-0.095=1*e^(-k*1)}}},  one year of decay time

{{{0.905=e^(-k)}}}

{{{ln(0.905)=ln(e^(-k))}}}

{{{-k=ln(0.905)}}}

{{{-k=-0.099820}}}

{{{k=0.099820}}}

Model can be stated  {{{highlight(P(t)=P[o]e^(-0.09982t))}}}.


FIND HALF-LIFE

Using model as  {{{1/2=1*e^(-0.09982t)}}}
take natural logs of both sides, simplify, and solve for t.

result should be {{{t[1/2]=ln(2)/0.09982}}}.