Step by step
The question
Question: graph y = log(x − 1).
A log with no small number is base 10.
A log is a power.
log of c is the power of 10 that makes c.
1 Find the domain
10 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 − 1.
So x − 1 > 0.
To remove the − 1, we add 1 on both sides.
So the domain is x > 1.
Books also write it as (1, ∞).
The round bracket means 1 itself is not in.
∞ means it goes on for ever.
2 Vertical asymptote
At the edge, the inside is 0: x = 1.
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 = 1.
3 Find some points
The log is easy when the inside is a power of 10: 1/10, 1 and 10.
We choose x to make that happen.
Make the inside 1/10: x is 11/10.
log of 1/10 is −1, because 10 to the power −1 is 1/10.
So y is −1.
Inside 1: x is 2.
log of 1 is 0, because 10 to the power 0 is 1.
So y is 0.
Inside 10: x is 11.
log of 10 is 1, because 10 to the power 1 is 10.
So y is 1.
4 Draw the graph
Draw the grid.
The asymptote x = 1, dashed.
Plot the points.
Join them with a smooth curve.
Near x = 1 it goes down along the dashed line, but never touches it.
So: domain x > 1, asymptote x = 1.
5 Check
Let's check: at x = 1 the inside is 0, and no power of 10 is 0.
So there is no point there: the asymptote is right.
The purple curve is y = log x, the plain log.
Ours is that curve moved right 1.
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.