In a city, the initial population is $$P(0)=70281$$ and 10 years after, $$P(10)=66277$$
I have to find $P(t)$ using the logistic model with max value equal to 80000, which gives the following equation $$\frac{1}{P}\cdot\frac{dP}{dt}=k\left(1-\frac{P}{80000}\right)$$
When I solved it, I found $$P(t)=\frac{80000}{1+Ae^{-kt}}$$ and from the inital conditions, $$A=0.1383,\qquad k=-0.04038$$
Hence, $$P(t)=\frac{80000}{1+0.1383e^{0.04038t}}$$
My question is: shouldn't $k>0$ and then, $$P(t)=\frac{80000}{1+0.1383e^{-0.04038t}}?$$
If so, what's wrong?
By the way, I plotted $P(t)$ using Geogebra and it makes sense for me, the population is decreasing. But I don't know if this could happen in the logistic model.