Step by step
The question
Question: solve log₅(x + 7) = log₅(3x − 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 Same log, so the insides are equal
Both sides are log base 5 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 5.
Now we have x + 7 = 3x − 1.
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 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.
3x − x = 2x.
Now we have 7 = 2x − 1.
Let's swap the two sides, so x is on the left.
We want x by itself.
Two things are stuck to x: times 2 and − 1.
The − 1 is on the outside, so it comes off first.
Then the times 2.
3 Remove the − 1
To remove − 1, we add 1.
We do it on both sides, so they stay equal.
− 1 and + 1 make zero.
Cross them out.
7 + 1 = 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 11: 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 − 1.
Then: remove the 2.
Last: check.