# How is this composite function calculated?

The problem arose from after reading the answer here:

How to adjust the parameters of Lotka-Volterra equations to fit the extremal values of each population

This is a composite function; $a= g(h(x_1/x_0))$

where $g(z) = z - 1 - \ln z$, $h(z) = \frac{\ln z}{z-1}$ and $x_0 = 200000$ and $x_1 = 800000$

The individual who provided the answer in the given link got $a = 3.2221 *10^4$

I get $0.23$ as the answer, which is obviously way off. I would really like to know how to solve this problem.

The individual in his answer (linked) implied the use of log to the base $e$ instead of simply using ln. This is what I was told anyways. Therefore, I used ln instead of log for the presentation of the problem here.

I am solving a similar problem and I need to understand how this is done.

• Compute $y=h(x_1/x_0)$ and then $g(y)$. – Math Lover Dec 28 '17 at 19:35
• @MathLover I did exactly that. so I compute h(4) right? and then g(h(4)) but my answer is always way off. – Selena Carlos Dec 28 '17 at 19:37
• I solved the problem as follows:h(x1/x0) h(x_1/x_0)= ln (4)/3 =0.46 and then I solved for g (0.46) = 0.46 -1 - ln(0.46)=0.23 – Selena Carlos Dec 28 '17 at 19:44

Straight forward calculation, step by step. First, $\frac{x_1}{x_0}= \frac{800000}{200000}= 4$. Second, $h(4)= \frac{ln(4)}{4- 1}= \frac{ln(4)}{3}= 0.46210$. Finally, $g(0.46210)= 0.46210- 1- ln(0.46210)= 0.46210- 1+ 0.7720= 0.2341$.