I have the following graph for a logarithmic function $f$:
I don't know any thing about writing an equation for a logarithmic function by knowing it's graph. All what I know is how to draw a graph for a logarithmic function equation.
How can I write an equation for the graph above or any another graph-logarithmic function graph-?
On
You pick a functional form with some number of parameters. "A logarithmic function" is not a sufficient description. Seeing the vertical asymptote, I would guess th form you want is $y=a \log (3-bx)$ To get the point $(-1,0)$, you need $b=\frac 34$. Now find the $a$ that fits best. If that doesn't satisfy, try another form. If it sounds like a bit of art, it is. You can use root-finders for the parameters once you choose a form.
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The equation must be undefined for x = 3 so log(x-3) will be part of the function. Use the first point (2, -2) to make the log term "log 1" as log((2-3)/-1). Use the second point to determine the base of the log, Log((1-3)/-1) is log 2 so the base is 2. Put it all together as y = -2 + log base 2 ((x-3)/-1).
Since you are told that you face a logarithmic function, you know by inspection that there is an infinite branch at x=3. Then, the function must be something like y = a + b Log[3-x]. Now, since you have two parameters (a and b), just use two of the data points of your graph; this leads to two linear equations you can simply solve. Just use the third data point to check that you reached the right answer. Can you continue with this ?