Step by step
The question
Question: solve log(x + 9) − log(x) = 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.
A log with no small number is base 10.
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: log((x + 9)/x).
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 10 to the power 1 is what is inside.
So (x + 9)/x is 10¹.
10¹ is 10.
3 Remove the fraction
To remove the fraction, we multiply by its bottom, x.
We do it on both sides, so they stay equal.
x divided by x is 1.
So just the top is left.
Now we have x + 9 = 10x.
4 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.
10x − x = 9x.
Now we have 9 = 9x.
Let's swap the two sides, so x is on the left.
5 Remove the 9
9x means 9 times x.
We want just x.
To remove times 9, we divide by 9.
We do it on both sides, so they stay equal.
9 divided by 9 is 1.
So just x is left.
9 divided by 9 is 1.
So x = 1!
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 = 1.
The insides are 10 and 1: both positive.
Divide them inside one log: log of 10 is 1.
It works!
So x = 1.
Let's look back.
First: make one log.
Then: write it as a power.
Then: remove the fraction.
Then: get x on one side.
Then: remove the 9.
Last: check.