Step by step
The question
Question: solve log(2x + 1) = log(x + 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.
A log with no small number is base 10.
1 Same log, so the insides are equal
Both sides are log 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 10.
Now we have 2x + 1 = x + 4.
2 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 right 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.
2x − x = x.
Now we have x + 1 = 4.
3 Remove the + 1
We want x by itself.
The + 1 is in the way.
To remove + 1, we take away 1.
We do it on both sides, so they stay equal.
+ 1 and − 1 make zero.
Cross them out.
4 − 1 = 3.
So x = 3!
4 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.
Both insides are 7: positive and the same.
It works!
So x = 3.
Let's look back.
First: same log, so the insides are equal.
Then: get x on one side.
Then: remove the + 1.
Last: check.