Graph y = -log_2(x - 1)

Step by step

The question

Question: graph y = −log₂(x − 1).

A log is a power.

log₂ of c is the power of 2 that makes c.

1 Find the domain

2 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 up and up.

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 2: 1/2, 1, 2 and 4.

We choose x to make that happen.

Make the inside 1/2: x is 3/2.

log₂ of 1/2 is −1, because 2 to the power −1 is 1/2.

So y is 1.

Inside 1: x is 2.

log₂ of 1 is 0, because 2 to the power 0 is 1.

So y is 0.

Inside 2: x is 3.

log₂ of 2 is 1, because 2 to the power 1 is 2.

So y is −1.

Inside 4: x is 5.

log₂ of 4 is 2, because 2 to the power 2 is 4.

So y is −2.

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 up 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 2 is 0.

So there is no point there: the asymptote is right.

The purple curve is y = log₂ x, the plain log.

Ours is that curve flipped upside down by the minus, and 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.