I'm doing a badly worded question on mathematical modelling and I'm close to finding the solution.
Find the time interval on which $M(t)\ge 0$ with dependence on $M_0$
Basically the $M_0$ is my initial population and it's dependent on $M_0 < \frac{5}{3}$. When $M_0 < \frac{5}{3}$ then $$(M_0 -\frac{5}{3})e^{12t}+\frac{5}{3}=0$$ will mean $M(t)\ge 0$
I need to solve this equation for $t$ (time interval) using logarithms but I can't seem to do it, without getting negative natural log. I'd appreciate some help with the algebra. Thanks :)
You can only solve it when $M_0 \lt \frac 53$ In that case you have $$\frac 53=\left(\frac 53-M_0\right)e^{12t}\\ \frac {\frac 53}{\frac 53-M_0}=e^{12t}\\ \log\left(\frac {\frac 53}{\frac 53-M_0}\right)=12t\\ \frac 1{12}\log\left(\frac {\frac 53}{\frac 53-M_0}\right)=t$$ and the argument of the log is positive, so no problem