Question

Determine the function which corresponds to the given graph.

a natural logarithmic function crossing the y axis at zero and going through the point 2,1.


The asymptote is x = -1.

Answer 1

The general natural logarithmic function $y=ln(x)$ has an asymptote at $x=0$. To make this function have an asymptote at $x=-1$, we need to translate the function 1 unit left. So the new function (according to rules of translations) takes the form

$y=ln(x+1)$

This function should also go through two points $(0,0)$ and $(2,1)$. Let's check (0,0).

$ln(0+1)=ln(1)=0$. So the first point is okay. Let's check (2,1).

$ln(2+1)=ln3$, which is not equal to 1.

A simple trick to make this equal to 1, for us to satisfy all the conditions of making this function, is to think "What can we do to $ln(3)$ to make it equal to 1?"

Simple! We have to multiply it by $\frac{1}{ln(3)}$!

Now, our final equation becomes $y=\frac{1}{ln(3)}ln(x+1)$.

ANSWER: $y=f(x)=\frac{1}{ln(3)}ln(x+1)$