Step by step
The question
Question: solve log₃(4x − 2) = log₃(2x + 6).
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 Same log, so the insides are equal
Both sides are log base 3 of something.
They are equal, so the insides are equal too.
Careful!
This only works when both logs have the same base.
Here both are base 3.
Now we have 4x − 2 = 2x + 6.
2 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the 2x from the right side.
To remove 2x, we take away 2x.
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.
2x and − 2x make zero.
Cross them out.
4x − 2x = 2x.
Now we have 2x − 2 = 6.
We want x by itself.
Two things are stuck to x: times 2 and − 2.
The − 2 is on the outside, so it comes off first.
Then the times 2.
3 Remove the − 2
To remove − 2, we add 2.
We do it on both sides, so they stay equal.
− 2 and + 2 make zero.
Cross them out.
6 + 2 = 8.
Now we have 2x = 8.
4 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
8 divided by 2 is 4.
So x = 4!
5 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 = 4.
Both insides are 14: positive and the same.
It works!
So x = 4.
Let's look back.
First: same log, so the insides are equal.
Then: get x on one side.
Then: remove the − 2.
Then: remove the 2.
Last: check.