How to find the Graph of a Logarithm from two Points

Question

I have a graph of a logarithm function with only two points marked. I need to find the equation for the function from these two points. The two points are $(8,3)$ and $(1,0)$. I already know the equation is $f(x)=\log_2(x)$. How can I find the equation from these two points and the rough shape of the graph of the function?

Answer 1

You may turn it into a problem to find a fitting exponential curve first which is a standard exercise in the context of exponential functions:

• $(8,3)$ and $(1,0)$ lie on a logarithmic curve $\Rightarrow$ $(3,8)$ and $(0,1)$ lie on an exponential curve.
• So, using $x = a\cdot b^y$, set up the equations $$8 = a \cdot b^3, 1 = a \cdot b^0 \Rightarrow a = 1, b= 2 \Rightarrow x = 2^y \Rightarrow y= \log_2 x$$

Answer 2

With only two data points you can determine at most two parameters for the equation of the curve, so you can locate the equation of the curve at best within a two-parameter family of curves.

For example, if you think the curve has equation

$y = \log_b(x) + c$

for some base $b$ and some constant $c$, then $y=0$ when $x=1$ tells you that $c=0$, and $y=3$ when $x=8$ tells you that $b^3=8$ and so $b=2$.

But if you think that the curve has equation

$y = \log_b(x) + cx + d$

for some base $b$ and constants $c$ and $d$ then two data points are not sufficient to determine the values of $b$, $c$ and $d$.

Answer 3

You need to define your function with the parameters of interest. It sounds like you have something like $f(x)=a+\log_b x$. You substitute in the points you have and solve the equations to find $a$ and $b$. We have $$0=a+\log_b 1\\3=a+\log_b 8\\0=a\\3=\log_b 8\\b^3=8\\b=2$$ Where I just got the last by inspection. If you don't notice that $2^3=8$ you can take the cube root of each side.