Step by step
The question
Question: graph y = log₃(x − 2).
A log is a power.
log₃ of c is the power of 3 that makes c.
1 Find the domain
3 to any power is always more than 0.
So a log only takes positive numbers.
The inside must be more than 0.
The inside is x − 2.
So x − 2 > 0.
To remove the − 2, we add 2 on both sides.
So the domain is x > 2.
Books also write it as (2, ∞).
The round bracket means 2 itself is not in.
∞ means it goes on for ever.
2 Vertical asymptote
At the edge, the inside is 0: x = 2.
Close to it, the graph goes down and down.
This is the vertical asymptote.
Careful!
The asymptote is where the inside is 0, not where x is 0.
Here that is x = 2.
3 Find some points
The log is easy when the inside is a power of 3: 1/3, 1, 3 and 9.
We choose x to make that happen.
Make the inside 1/3: x is 7/3.
log₃ of 1/3 is −1, because 3 to the power −1 is 1/3.
So y is −1.
Inside 1: x is 3.
log₃ of 1 is 0, because 3 to the power 0 is 1.
So y is 0.
Inside 3: x is 5.
log₃ of 3 is 1, because 3 to the power 1 is 3.
So y is 1.
Inside 9: x is 11.
log₃ of 9 is 2, because 3 to the power 2 is 9.
So y is 2.
4 Draw the graph
Draw the grid.
The asymptote x = 2, dashed.
Plot the points.
Join them with a smooth curve.
Near x = 2 it goes down along the dashed line, but never touches it.
So: domain x > 2, asymptote x = 2.
5 Check
Let's check: at x = 2 the inside is 0, and no power of 3 is 0.
So there is no point there: the asymptote is right.
The purple curve is y = log₃ x, the plain log.
Ours is that curve moved right 2.
The same shape.
It works!
Let's look back.
First: find the domain.
Then: vertical asymptote.
Then: find some points.
Then: draw the graph.
Last: check.