Logarithmic function that outputs [0.25, 1] when $x$ is [1,$n$]

Question

I'm looking for a formula that produces a logarithmic curve that meets the following conditions:

Given a set of positive integers from 1 to $n$, produce a logarithmic function that outputs a $y$ value of 0.25 when $x$ = 1 and the a $y$ value of 1 when $x$ = $n$. I'd also like to control the curvature of the logarithmic term.

Note: I don't have a maths background so may not be using the correct terms.

Answer 1

First you can try to think of a logarithmic function that outputs $[0, 0.75]$ on $[1,n]$, you just need to add the constant $+0.25$ to correct it afterwards.

With this setup we see that $\ln(x) = 0$ for $x=1$ so the lower part of the interval is already right. Now we want to correct the logarithm so that $\ln(n) = 0.75$. For that we can add a constant factor that would scale the logarithm like $a\ln(x)$. This would also make sure that we don't influence the value for when $x=1$ since this would still be $0$ whatever $a$ is.

Now we just need to solve $a\ln(n) = 0.75$ and get $a = 0.75/\ln(n)$.

I let you put all the parts together.