Solve log_3(x + 8) - log_3(x) = 2

Step by step

The question

Question: solve log₃(x + 8) − log₃(x) = 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: log₃((x + 8)/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 3 to the power 2 is what is inside.

So (x + 8)/x is 3².

3² is 9.

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 + 8 = 9x.

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.

9x − x = 8x.

Now we have 8 = 8x.

Let's swap the two sides, so x is on the left.

5 Remove the 8

8x means 8 times x.

We want just x.

To remove times 8, we divide by 8.

We do it on both sides, so they stay equal.

8 divided by 8 is 1.

So just x is left.

8 divided by 8 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 9 and 1: both positive.

Divide them inside one log: log base 3 of 9 is 2.

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 8.

Last: check.