Question 574650
You are trying to find a function where variable y depends on the value of variable x. What you need to find is values for a, b, h, and k. I would call a, b, h, and k parameters, because once we find them, they will be constants, while x and y will still be variables. Different values of those parameters will give different functions, but all those functions are part of the same family of functions.
You could say that {{{y=log(10,x)}}} is the mother of all the logarithmic functions. Or you could prefer to say that {{{y=ln(x)}}} is the mother of them all. Or maybe you would like another base. Here are the graphs of {{{y=log(1.6,x)}}} (red), and {{{y=log(2,x)}}} (green):
{{{graph(200,200,-1,9,-5,5,log(1.6,x),log(2,x))}}}
The graph of {{{y=a*log(b,x)}}} looks like that. No matter what values are chosen for a and b, the function only exists for x>0 and has x=0 for a vertical asymptote.
The graph of function {{{y=a*log(b,(x-h))+k}}} is similar, but moved h units to the right and k units up. It only exists for x-h>0 (x>h) and its vertical asymptote would be x-h=0 or x=h. That's what happened to your function: h=2.
Now that you have {{{y=a*log(b,(x-2))+k}}}, you can use the points given to find k (and a and b).
Point (3,5) says that {{{5=a*log(b,(3-2))+k}}} --> {{{5=a*log(b,1)+k}}} --> {{{5=a*0+k}}} --> {{{5=k}}}
Now that you have {{{y=a*log(b,(x-2))+5}}}, you can use point (5,9) to find a and b.
{{{9=a*log(b,(5-2))+5}}} --> {{{9=a*log(b,3)+5}}} --> {{{4=a*log(b,3)}}} --> {{{4=a(1/log(3,b))}}} --> {{{a=4*log(3,b)}}}
That is all you can do. You could chose b=3, which would make a=4, and your solution would be
{{{highlight(y=4*log(3,(x-2))+5)}}}
You could also chose b=9, which would make a=8, or infinite other choices. The function would be the same, just written in a different, equivalent form. It's just like 4/10 and 0.4 are different ways of writing 2/5, but it is still the same number.
Parameters a and b are related because the change of base is just a change of scale.
{{{8*log(9,(x-2))+5=8*(log(3,(x-2))/log(3,9))+5=8*(log(3,(x-2))/2)+5=4*log(3,(x-2))+5)}}}