Question 1202012
.
If 80​% of a radioactive element remains radioactive after 200 million​ years, 
then what percent remains radioactive after 700 million​ years? What is the half life of this element?
~~~~~~~~~~~~~~~~~~~~~~~~


<pre>
Let T be the half-life, in millions years.


We are given that

    0.8 = {{{(1/2)^(200/T)}}}.


To find the half-life from here, take the logarithm base 10 of both sides.  You will get

    log(0.8) = {{{(200/T)*log((0.5))}}},    {{{log((0.8))/log((0.5))}}} = {{{200/T}}}

    T = {{{(200*log((0.5)))/log((0.8))}}} = 621.2567  million years (the half-life).


The percent remaining after 700 millions years from the beginning is

    {{{(1/2)^(700/621.2567)}}} = 0.45795 = 45.795%  (rounded).    
</pre>

Solved.


------------


On radioactive decay, &nbsp;see the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/logarithm/Radioactive-decay-problems.lesson>Radioactive decay problems</A> 

in this site.


You will find many similar &nbsp;(and different) &nbsp;solved problems there.



/////////////////



The difference between my solution and the &nbsp;@Theo' solution is that &nbsp;@Theo 
makes &nbsp;TONS &nbsp;of unnecessary calculations on the way, &nbsp;while I make &nbsp;NO &nbsp;ONE &nbsp;unnecessary calculation.


The method which I use (with the half-life decay formula) is ALWAYS preferable,
when half-life is given or half-life is under the question.


It is even not a subject to discuss - - - it is the way to follow.