Step by step
The question
Question: solve log₃(x + 1) + log₃(x − 1) = 1.
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: log₃((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 3 to the power 1 is what is inside.
So (x + 1)(x − 1) is 3¹.
3¹ is 3.
3 Open the brackets
Each part times each part, then like terms: x² − 1 = 3.
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.
3 + 1 = 4.
Now we have x² = 4.
5 Square root of both sides
x squared is 4.
So x is the square root of 4, or minus it.
Minus times minus is plus, so the negative one works too.
2 times 2 is 4.
So x = 2 or x = −2.
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 = −2.
Inside a log we get −1.
There is no log of a negative number.
So x = −2 is a false answer.
Books call it extraneous.
Now x = 2.
The insides are 3 and 1: both positive.
Times them inside one log: log base 3 of 3 is 1.
It works!
Our answers were:
We cross out the false answer.
Only x = 2 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.