Solve log_2(x + 6) - log_2(x) = 2

Step by step

The question

Question: solve log₂(x + 6) − log₂(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: log₂((x + 6)/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 2 to the power 2 is what is inside.

So (x + 6)/x is 2².

2² is 4.

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 + 6 = 4x.

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.

4x − x = 3x.

Now we have 6 = 3x.

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

5 Remove the 3

3x means 3 times x.

We want just x.

To remove times 3, we divide by 3.

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

3 divided by 3 is 1.

So just x is left.

6 divided by 3 is 2.

So x = 2!

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 = 2.

The insides are 8 and 2: both positive.

Divide them inside one log: log base 2 of 4 is 2.

It works!

So x = 2.

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 3.

Last: check.