functionsgraphing-functionslogarithms
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-?
If you take the log of all your values, you can then treat it just like a linear scale. Using base 10 logs, if your data ranges from 10 to 100,000, taking logs makes it range from 1 to 5. Then scale that as you have before, presumably giving 10 lines to each unit to make 40 lines in your plot. Then plot you * where the log of the value belongs, so if you have a value of 2,000, the log of that is 3.3 and you would plot it on the 24th line.
So $y = e^{2x} $ is a function: it takes values of $x$ as input, and outputs the function value $y = e^{2x}$.
Here, we have that the interval $[0, \ln 2]$ defines the domain of $x$, for the purposes of your graph.
$\ln 2 \in \mathbb R$ is a constant, and defines an endpoint of the domain in question.
Note that $y = e^{2x}$ at $x = \ln 2$ outputs $ e^{\large 2\cdot \ln 2} = e^{\large \ln(2^2)} = e^{\large \ln 4} = 4$.
So you can plot the "end points" of your graph: $(0, e^{2\cdot 0}) = (0, 1)$, and the point $(\underbrace{\ln 2}_{x}, \;\underbrace{e^{2\cdot \ln 2}}_{y}) = (\ln 2, 4)$.
Note that $\ln 2 \approx .6931$.
We also have the function $f(y) = \ln y$, and in this case, we have that the function depends on the input value of $y$, which are all real values greater than zero. So here, we consider the ordered pairs $(y, \ln y)$, and obtain the plot (blue graph):
For a handy on-line graphing resource, feel free to explore with Wolfram Alpha, which I used to obtain the graphs posted above, and also use as a calculator. It is quite powerful, for you purposes!
You'll also want to start developing some intuition on the shapes of the most important functions, two of which are $f(x) = e^x$ and $f(x) = \ln x$. The best way to develop such an intuition is through practice: graphing by hand, and graphing using a graphing calculator and/or an on-line resource! Good luck!
Best Answer
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.