I'm basically trying to recreate the graph picture below. Using a online graphing tool "Desmos":
I managed to create the equations for the straight lines and circles for the sunset picture.
However, I tried creating the equations for the two curved lines, but I am having no luck.. I am completely stuck :( The answer I got trying to solve for the first curve line(green) is y=3 log3(-27x). I don't understand why its so off.
Any help would be greatly appreciated!
*Edit: First curve line is y=5+4log3(−x). (Working on restricting the points at the moment)
The original picture tells you the curve has the form $y=a+c\log_b(dx)$ for some constants $a$, $b$, $c$, and $d$, and that it passes through the points $(-27,17)$, $(-9,13)$, and $(-3,9)$. Since $27$, $9$, and $3$ are powers of $3$, it makes sense to try $b=3$ for the base of the logarithm (as you did). If you let $d=-1$ (it has to be negative because your $x$'s are negative and $\log(dx)$ is defined only when $dx$ is positive), then you get
$$\begin{align} 17&=a+3c\\ 13&=a+2c\\ 9&=a+c\\ \end{align}$$
If you pick any two of these equations and solve for $a$ and $c$, you get $a=5$ and $c=4$. (For example, subtracting the second equation from the first gives $17-13=(a+3c)-(a+2c)$, which reduces to $4=c$. Note, there are three equations here in only two unknowns, so you have to check that the answer really does satisfy all three equations, but of course it does.)