Step by step
The question
Question: graph y = log₃ x.
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.
So x > 0.
So the domain is x > 0.
Books also write it as (0, ∞).
The round bracket means 0 itself is not in.
∞ means it goes on for ever.
2 Vertical asymptote
At the edge, the inside is 0: x = 0.
Close to it, the graph goes down and down.
This is the vertical asymptote.
Careful!
There is no log of 0, and no log of a negative number.
No power of 3 gives them.
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: log₃ of 1/3 is −1, because 3 to the power −1 is 1/3.
So y is −1.
Inside 1: log₃ of 1 is 0, because 3 to the power 0 is 1.
So y is 0.
Inside 3: log₃ of 3 is 1, because 3 to the power 1 is 3.
So y is 1.
Inside 9: 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 = 0, dashed.
Plot the points.
Join them with a smooth curve.
Near x = 0 it goes down along the dashed line, but never touches it.
So: domain x > 0, asymptote x = 0.
5 Check
Let's check: at x = 0 the inside is 0, and no power of 3 is 0.
So there is no point there: the asymptote is right.
Let's look back.
First: find the domain.
Then: vertical asymptote.
Then: find some points.
Then: draw the graph.
Last: check.