# Differential equation for bacterial growth

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:

$$y=y_0*e^{kt}$$

However, my solution ended up being modelled as :

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

The resulting equation was:

$$y=e^{kt}*e^{ln500}$$

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.

Thanks.

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

## 1 Answer

$$e^{\ln 500}$$ 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 $$y = e^x$$.