Gompertz Growth Model problem with an equation of $P=a\ln(\frac{k}{P})$

Question

I'm having a trouble to solve the problem. Given: $P_1=0.03274448$ $P_1=0.03253040$. Since $P=a\ln\frac{K}{P}$ then, $$\frac{a\ln(\frac{K}{26.273})}{a\ln(\frac{K}{27.165})}=\frac{0.03274448}{0.03253040}$$ Find the values of $K$ and $a$

Answer 1

Let's start with $$ \begin{cases} a\ln \frac{K}{26.273} &= 0.03274448 \\ a \ln \frac{K}{27.165} & = 0.03253040 \end{cases} $$ when you consider $a\neq 0$ and divide the equations with each other, you get $$ \frac{ \ln \frac{K}{26.273} }{ \ln \frac{K}{27.165} } = \frac{0.03274448}{0.03253040} \approx 1.00658 = C $$ (I rewrote the result of the division as constant $C$ just to simplify the following algebra.) Now it's possible to solve for $K$. Let's first use the laws of logarithms like so: $$ \frac{ \ln \frac{K}{26.273} }{ \ln \frac{K}{27.165} } = \frac{\ln K - \ln 26.273}{\ln K - \ln 27.165} = C $$ now multiply both sides by the denominator, to get $$ \ln K - \ln{26.273} = C\left( \ln K - \ln{27.165} \right) = C \ln K - C\ln{27.165} $$ Now let's move all the terms involving $K$ to the left side, and the terms not involving $K$ to the right side: $$ \ln K - C \ln K= \ln 26.273 - C \ln 27.165 $$ $$ (1 - C) \ln K= \ln 26.273 - C \ln 27.165 $$ $$ \ln K= \frac{\ln 26.273 - C \ln 27.165}{1-C} $$ $$ K = \exp \left( \frac{\ln 26.273 - C \ln 27.165}{1-C} \right) $$ Now you only need to plug in the known values to this expression to get the value of $K$.

How about $a$ ? Turn your attention back to the first equation, and take either the first or the second equation. You know the value of $K$, so you can easily solve for $a$ by dividing out the $\ln{}$-term.

Is it clear now? Any questions?