I am assigned with a question which states the rate of a microbial growth is exponential at a rate of (15/100) per hour. where y(0)=500, how many will there be in 15 hours?

I know this question is generally modelled as:


However, my solution ended up being modelled as :

$y=e^{kt}*e^{c}$ via $y'=ky$

The resulting equation was:


I ended up getting $y(15)=4743.86$ which is the same answer for both methods.

I'm wondering how the general equation was modelled, and if someone could explain how I could tidy up my equations.


  • $\begingroup$ You mean y'=ky instead of y'=kx $\endgroup$ – dallonsi Aug 4 at 20:33
  • $\begingroup$ Is there a typo? Should be $y'=ky,$ because in the exponential growth is the speed $y'$ of the growth proportional to the amount $y.$ $\endgroup$ – user376343 Aug 4 at 20:33
  • $\begingroup$ Yes, oops, my bad $\endgroup$ – persimonns Aug 4 at 20:34
  • 1
    $\begingroup$ My end value would be $500\cdot 1.15^{15}=4068.53$. $\endgroup$ – Christian Blatter Aug 5 at 8:13

$e^{\ln 500}$ is equal to 500. Notice that if $y = e^x$, taking the natural log of both sides gives you $\ln y = \ln (e^x) = x$. Thus, to undo this operation, take each side as the power of $e$ to get $e^{\ln y} = e^x$, which must be $y = e^x$.


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