Question 1155415


{{{y=4e^(-0.5t) }}}

 in the form {{{y=a(1+r)^t}}} or{{{ y=a(1-r)^t }}}


first you need to find {{{r}}} that makes following statement true


{{{e^(-0.5t) =(1+r)^t}}}.....take log of both sides


{{{log(e^(-0.5t) )=log((1+r)^t)}}}


{{{(-0.5t)log(e )=t*log((1+r))}}}


{{{(-0.5cross(t))log(e )/cross(t)=log((1+r))}}}...write {{{-0.5}}} as {{{-1/2}}}


{{{(-1/2)log(e )=log((1+r))}}}


{{{log(e^(-1/2) )=log((1+r))}}}.....apply exponent rule {{{e^(-1/2)=1/e^(1/2)}}}


{{{log(1/e^(1/2) )=log((1+r))}}}.......{{{e^(1/2)=sqrt(e)}}}


{{{log(1/sqrt(e) )=log((1+r))}}}.....if log same, we have


{{{1/sqrt(e) =1+r}}}


{{{r = 1/sqrt(e) - 1}}}


your equation in the form {{{y=a(1+r)^t}}} is:
 

{{{a=4}}}, {{{r = 1/sqrt(e) - 1}}}}


{{{y=4(1+1/sqrt(e) - 1)^t}}}


{{{y=4(1/sqrt(e) )^t}}}


check:

{{{ 4e^(-0.5t)=4(1/sqrt(e) )^t }}}-> true


or 

{{{y=4(1-r)^t}}}

{{{y=4(1-(1/sqrt(e) - 1))^t}}}

{{{y=4(1-1/sqrt(e)+ 1)^t}}}

{{{y=4(2-1/sqrt(e))^t}}}

check:

{{{ 4e^(-0.5t)=4(2-1/sqrt(e))^t}}}-> not true


so, your answer is: {{{y=4(1/sqrt(e) )^t}}}