Step by step
The question
Question: solve log₂(x − 1) + log₂(x + 1) = 3.
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 − 1)(x + 1)).
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 3 is what is inside.
So (x − 1)(x + 1) is 2³.
2³ is 8.
3 Open the brackets
Each part times each part, then like terms: x² − 1 = 8.
4 Get x² alone
There is no x term, so we get x² by itself.
To remove the − 1, we add 1 on both sides.
8 + 1 = 9.
Now we have x² = 9.
5 Square root of both sides
x squared is 9.
So x is the square root of 9, or minus it.
Minus times minus is plus, so the negative one works too.
3 times 3 is 9.
So x = 3 or x = −3.
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 = −3.
Inside a log we get −4.
There is no log of a negative number.
So x = −3 is a false answer.
Books call it extraneous.
Now x = 3.
The insides are 2 and 4: both positive.
Times them inside one log: log base 2 of 8 is 3.
It works!
Our answers were:
We cross out the false answer.
Only x = 3 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.