In the function $f(x)=k\ln(ax+b)$ what does the $k$ value represent? Basically what the title of the question is, in the function $f(x)=k\ln(ax+b)$ I presume that $a$ is the slope, $k\ln(b)$ is the y intercept but what is $k$?
 A: $k, a$ and $b$ are just constants, which you might choose. For example you could set $k=1$, $a=2$ and $b=10$. The question is to find the values of these parameters, so you get the pictured graph.
You can see for example that the point (0|0) lies on the graph.
Hence you have $f(0)=0$, which translates to $k\ln(a\cdot 0+b)=0$.
So $k\ln(b)=0$. Can you deduce the value of $b$?
Do you find more information in the pictured graph to deduce the other values?
You might want to check a function plotter, like geogebra:
https://www.geogebra.org/calculator
Plug in the expression k * ln(a * x +b)
There you will be able to change the values for $k$, $a$ and $b$, and you will see how this affects the graph.
A: The graph of $y-k=f(x-h)$ simply translates the graph of $y=f(x)$ so that $(h,k)$ is the new origin (e.g. if $h,k>0$, this would mean translating $y=f(x)$ to the right by $h$ units and up by $k$ units).
The graph of $y=Af(x)$ vertically stretches the graph of $y=f(x)$ by a factor of $A$ (note that if $A<0$, the graph is reflected over the $x$-axis and stretched by a factor of $|A|$).
Your example has both of these transformations for $a\neq 0$ applied to $y=\ln x$:
$$y=k\ln (ax+b)\implies y-k\ln(a)=k\ln (x-(-b/a)).$$
As for finding what $k,a,b$ are in your example, you can do so if you know three points on your function, from which you can set up three equations and solve for these parameters.
