I am given two points $(1.5, 3.1)$ and $(6.5, 1.45)$ and I have to find a function that cuts through these points. I have chosen to use an $\ln(x)$ function in the general form:
$y=a\ln(x-h)-k$
I know that an additional point is needed to find the values of $a$, $h$ and $k$ however i want to write $h$ and $k$ in terms of $a$. Then i can simply test different values of $a$ to see which one fits my needs.
When you have the value of $h$ finding the value for $k$ is easy:
$k=a\ln(x_1-h)-y_1$
So the question is: How to find $h$ when $a$ is given?
You have two equations:
$y_1=a\ln(x_1-h)-k\\ y_2=a\ln(x_2-h)-k$
($(x_1,y_1)$ and $(x_2,y_2)$ are the two given points.)
Subtract the two equations from each other, divide both sides by $a$ and you'll get:
$\frac{y_1-y_2}a=\ln(x_1-h)-\ln(x_2-h)$
You may now exponentiate both sides of the equation:
$e^{\frac{y_1-y_2}a}=\frac{x_1-h}{x_2-h}$
... which will lead to a rather complex term for $h$ which is depending on $a$.
If I did not make a mistake it is:
$h = \frac{x_1-x_2e^{\frac{y_1-y_2}a}}{1-e^{\frac{y_1-y_2}a}}$
Then you can insert that term into the term $k=a\ln(x_1-h)-y_1$ ...
Note that not all values of $\mathbb R$ are allowed for $a$!