Question 253874
your equation to work with is:


{{{C[t]=C[0]*2*e^(-t/h)}}}


The exponent in the equation is {{{-t/h}}}.


{{{C[0] = 240}}}
{{{C[t] = 7.5}}}
{{{h = 13.8}}}


substituting in that equation, you get:


{{{7.5 = 240 * 2 * e^(-t/13.8)}}}


the exponent in the equation is {{{-t/13.8)}}}


you want to isolate the exponential term on the right hand side of the equation so you need to divide both sides of the equation by 240 * 2 to get:


{{{7.5/(2*240) = e^(-t/13.8)}}}


the exponent in the equation is {{{-t/13.8)}}}


take the log of both sides of this equation to get:


{{{log(7.5/(2*240)) = log(e^(-t/13.8))}}}


since log(b^c) = c*log(b), your equation becomes:


{{{log(7.5/(2*240)) = (-t/13.8) * log(e)}}}


multiply both sides of this equation by 13.8 to get:


{{{13.8*log(7.5/(2*240)) = -t*log(e)}}}


divide both sides of this equation by -log(e) to get:


{{{t = (13.8*log(7.5/(2*240)))/(-log(e))}}}


solve for t to get:


{{{t = -24.92528364/-.434294482 = 57.39258655}}} seconds.


substitute in original equation to see if this value for t is good.


original equation is:


{{{C[t]=C[0]*2*e^(-t/h)}}}


exponent in the equation is {{{-t/h}}}


{{{C[0] = 240}}}
{{{C[t] = 7.5}}}
{{{h = 13.8}}}
t = 57.39258655


equation becomes:



{{{7.5=240*2*e^(-57.39258655/13.8)}}}


exponent in the equation is {{{(-57.39258655/13.8)}}}


simplify to get:


{{{7.5=480*e^(-4.158883083)}}}


simplify further to get:


{{{7.5 = 7.5}}}


since the equation is true, the value for t is good.


t = 57.39258655 seconds