How do I go about finding the $log$ function that passes through 2 points?

Question

In my example I have a point $P$:

$P = (1,3)$

And another point $Q$

$Q = (8,8)$

I need to find a logarithmic function which passes through both $P$ and $Q$ and

$y = 8$ when $x > 8$.

I have no idea where to start. How can I go about finding such a function?

Answer 1

If you want a curve of the form $y=a\log x + b$ you can just plug your two points in to find $a$ and $b$. We get $$3=a \log 1 +b\\8=a \log 8 +b\\3=b\\5=a\log 8\\a=\frac 5{\log 8}\\y=\frac 5{\log 8}\log x+3$$ The derivation does not care what base of logs you use. If you use $2$ the final expression simplifies to $$y=\frac 53\log_2 x +3$$

Answer 2

Seeing one of the points is $(1,3)$ then solve the equation $a\cdot ln(8) + 3 = 8$

$a = \frac{5}{ln(8)} = 2.4045$

Hence $y = 2.4045\cdot ln(x) + 3$