Step by step
The question
Question: solve log₂(x + 3) + log₂(x − 3) = 4.
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 + 3)(x − 3)).
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 4 is what is inside.
So (x + 3)(x − 3) is 2⁴.
2⁴ is 16.
3 Open the brackets
Each part times each part, then like terms: x² − 9 = 16.
4 Get x² alone
There is no x term, so we get x² by itself.
To remove the − 9, we add 9 on both sides.
16 + 9 = 25.
Now we have x² = 25.
5 Square root of both sides
x squared is 25.
So x is the square root of 25, or minus it.
Minus times minus is plus, so the negative one works too.
5 times 5 is 25.
So x = 5 or x = −5.
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 = −5.
Inside a log we get −2.
There is no log of a negative number.
So x = −5 is a false answer.
Books call it extraneous.
Now x = 5.
The insides are 8 and 2: both positive.
Times them inside one log: log base 2 of 16 is 4.
It works!
Our answers were:
We cross out the false answer.
Only x = 5 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.