Question 576524
{{{A=A[0](1/2)^(t/T)}}} is the equation given and you want the value of {{{t}}} that will make {{{A/A[0]=2/5}}}
(You know it has to be more than {{{T=3.9 * 10^9years}}} because at T you would have 1/2 left and 2/5<1/2).
We could divide both sides by {{{A[0]}}} to get
{{{A/A[0]=(1/2)^(t/T)}}}
Taking logarithm of both sides
{{{log((A/A[0]))=(t/T)*log((1/2))}}} --> {{{log((A/A[0]))=(t/T)*(log(1)-log(2))}}} --> {{{log((A/A[0]))=(t/T)*(0-log(2))}}} --> {{{log((A/A[0]))=(t/T)*(-log(2))}}} --> {{{log((A/A[0]))=-(t/T)*log(2)}}}
Plugging in the values given,
{{{log((2/5))=-(t/(3.9 * 10^9))*log(2)}}}
Using the approximate values for the logarithms
{{{-0.39794=-0.30103*t/(3.9 * 10^9)}}}
So, {{{t=0.39794*(3.9* 10^9)/0.30103}}}
{{{highlight(t=5.16*10^9years)}}}