Solve log_2(x + 2) + log_2(x - 2) = 5

Step by step

The question

Question: solve log₂(x + 2) + log₂(x − 2) = 5.

One rule first: what is inside a log must be positive.

There is no log of 0 or of a negative number.

We use it in the check.

1 Make one log

The product rule, from the log rules lesson: plus outside the logs is times inside one log.

So we get one log: log₂((x + 2)(x − 2)).

Careful!

Plus between two logs is not plus inside.

It is times inside one log.

2 Write it as a power

A log is a power.

So 2 to the power 5 is what is inside.

So (x + 2)(x − 2) is 2⁵.

2⁵ is 32.

3 Open the brackets

Each part times each part, then like terms: x² − 4 = 32.

4 Get x² alone

There is no x term, so we get x² by itself.

To remove the − 4, we add 4 on both sides.

32 + 4 = 36.

Now we have x² = 36.

5 Square root of both sides

x squared is 36.

So x is the square root of 36, or minus it.

Minus times minus is plus, so the negative one works too.

6 times 6 is 36.

So x = 6 or x = −6.

6 Check

An answer can be false here: a log of a negative number does not exist.

So we check every answer in the question.

Let's check x = −6.

Inside a log we get −4.

There is no log of a negative number.

So x = −6 is a false answer.

Books call it extraneous.

Now x = 6.

The insides are 8 and 4: both positive.

Times them inside one log: log base 2 of 32 is 5.

It works!

Our answers were:

We cross out the false answer.

Only x = 6 is left.

Let's look back.

First: make one log.

Then: write it as a power.

Then: open the brackets.

Then: get x² alone.

Then: square root of both sides.

Last: check.