[Math] Creating a function with logarithmic growth

Question

I have some knobs with an internal value of $0$ to $1$. These represent a value in a range, like $1$ to $1000$.
Case in point, I would like to be able to change the scale/growth of the display value. For instance, the display value with linear growth:

ling(x) =  min + (max-min) * x

Where $x$ is between $0$ and $1$.
Similarly with exponential growth:

expg(x) = min * (max/min)^x

Is there a similar rule/formula with logarithmic properties?

edit:
Okay i've been trying out some different things. Originally i worked with this:

logg(x) = (max - min)/log(max - min + 1) * log((max - min) * x + 1) + min

But i realized the slope was not the inverse of the expg function (which should be an identity of the logarithmic function?). I decided to mirror the expg(x) function instead:

lelogg(x) = ling(1 - x) - expg(1 - x) + ling(x)

which seems perfect:

but it begs the question, which of these graphs has true logarithmic / exponential growth?

Answer 1

ling(x) is a linear function as it corresponds to $y=a+bx$ if you set min=a and max=a+b

expg(x) is an exponential function as it corresponds to $y=ae^{bx}$ if you set min=a and max=a*exp(b)

logg(x) is almost a logarithmic function of the form $y=a\log(x)+b$ except that you have log((max - min) * x + 1) when log((max - min) * x) would be better, and in general the whole expression could be simpler

lelogg(x) is not a logarithmic function, but instead the difference between a constant and a negative exponential function, so is bounded above, unlike a logarithmic function. Note that ling(1 - x) + ling(x)is the same as max+min