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?

Impossible for a logarithmic function to pass trough $(1, 3)$ — Jakobian, Jun 04 '18 at 17:27
What is the form of the function you are looking for? Is it $y=a \log x$, for example? That has only one parameter, so is unlikely to pass through two points. It also has the problem that $\log 1=0$ — Ross Millikan, Jun 04 '18 at 17:29
rather than a logarithmic function, you could try polynomials — The Integrator, Jun 04 '18 at 17:32
@RossMillikan Rather than having a linear line between $P$ and $Q$, I want to have a function which looks concave-down and increasing between those two points, and the inverse of an exponential increase in $y$ between those 2 points.(EDIT: concave-down not up) — Andrew Kor, Jun 04 '18 at 17:36
You should have said that at the beginning. You can follow the approach in my answer to evaluate $k$ and $c$ — Ross Millikan, Jun 04 '18 at 17:45

score 3 · Accepted Answer · answered Jun 04 '18 at 17:42

3

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 $$y=\frac 53\log_2 x +3$$

answered Jun 04 '18 at 17:42

Ross Millikan

374,822

Sorry my vagueness, my question was not specific enough. I need an example of a function that's the inverse of an exponential function, which starts at $P$ and passes through $Q$. However when $x>8$ then $y = 8$. So a function that is inverse of exponential and then stops increasing after $Q$ – Andrew Kor Jun 04 '18 at 17:54
To stop it increasing after $x=8$ you can just declare that $y=8$ for all $x \ge 8$. – Ross Millikan Jun 04 '18 at 17:59

score 0 · Answer 2 · answered Jun 04 '18 at 17:52

0

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

answered Jun 04 '18 at 17:52

Phil H

5,579

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

2 Answers2