Step by step
The question
Question: solve log₂(x + 9) − log₂(x − 3) = 2.
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 quotient rule, from the log rules lesson: minus outside the logs is divide inside one log.
So we get one log: log₂((x + 9)/(x − 3)).
Careful!
Minus between two logs is not minus inside.
It is divide inside one log.
2 Write it as a power
A log is a power.
So 2 to the power 2 is what is inside.
So (x + 9)/(x − 3) is 2².
2² is 4.
3 Remove the fraction
To remove the fraction, we multiply by its bottom, x − 3.
We do it on both sides, so they stay equal.
(x − 3) divided by (x − 3) is 1.
So just the top is left.
Now we have x + 9 = 4(x − 3).
4 Open the brackets
The 4 in front of the bracket multiplies every part inside.
So we write 4 times each part.
Work out each multiplication.
5 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the x from the left side.
To remove x, we take away x.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
x and − x make zero.
Cross them out.
4x − x = 3x.
Now we have 9 = 3x − 12.
Let's swap the two sides, so x is on the left.
We want x by itself.
Two things are stuck to x: times 3 and − 12.
The − 12 is on the outside, so it comes off first.
Then the times 3.
6 Remove the − 12
To remove − 12, we add 12.
We do it on both sides, so they stay equal.
− 12 and + 12 make zero.
Cross them out.
9 + 12 = 21.
Now we have 3x = 21.
7 Remove the 3
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just x is left.
21 divided by 3 is 7.
So x = 7!
8 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 = 7.
The insides are 16 and 4: both positive.
Divide them inside one log: log base 2 of 4 is 2.
It works!
So x = 7.
Let's look back.
First: make one log.
Then: write it as a power.
Then: remove the fraction.
Then: open the brackets.
Then: get x on one side.
Then: remove the − 12.
Then: remove the 3.
Last: check.