Solve log_2(3x - 1) = log_2(x + 5) Easy

The question

Question: solve log₂(3x − 1) = log₂(x + 5).

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 2 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 2.

Now we have 3x − 1 = x + 5.

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.

3x − x = 2x.

Now we have 2x − 1 = 5.

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.

5 + 1 = 6.

Now we have 2x = 6.

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.

6 divided by 2 is 3.

So x = 3!

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 = 3.

Both insides are 8: 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.

Then: remove the 2.

Last: check.

▶ Watch it solved

Solve log_2(x) + log_2(x - 2) = 3 Medium

The question

Question: solve log₂(x) + log₂(x − 2) = 3.

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 product rule, from the log rules lesson: plus outside the logs is times inside one log.

So we get one log: log₂(x(x − 2)).

Careful!

Plus between two logs is not plus inside.

It is times inside one log.

2 Write it as a power

A log is a power.

So 2 to the power 3 is what is inside.

So x(x − 2) is 2³.

2³ is 8.

3 Open the brackets

x times each part, then like terms: x² − 2x = 8.

4 Get 0 on one side

To solve it, we want 0 on the right side.

So we take away 8 on both sides.

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

Now we have x² − 2x − 8 = 0.

5 Factor

We need two numbers.

They multiply to −8, and they add to −2.

They multiply to a minus number, so one number is plus and one is minus.

They add to a minus number, so the bigger one is minus.

We list the pairs that make 8, smallest first.

We take the smaller from the bigger.

We need 2.

2 and 4 give 2.

Yes!

With the signs: 2 and −4.

So it is (x + 2)(x − 4) = 0.

6 One bracket is 0

Two numbers multiply to 0.

That only happens if one of them is 0.

So one bracket must be 0.

We make each bracket 0.

Two small equations.

In x + 2 = 0, we take away 2 on both sides.

In x − 4 = 0, we add 4 on both sides.

So x = −2 or x = 4.

7 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.

Inside a log we get −2.

There is no log of a negative number.

So x = −2 is a false answer.

Books call it extraneous.

Now x = 4.

The insides are 4 and 2: both positive.

Times them inside one log: log base 2 of 8 is 3.

It works!

Our answers were:

We cross out the false answer.

Only x = 4 is left.

Let's look back.

First: make one log.

Then: write it as a power.

Then: open the brackets.

Then: get 0 on one side.

Then: factor.

Then: one bracket is 0.

Last: check.

▶ Watch it solved

Solve log_2(x - 1) + log_2(x + 1) = 3 Harder

The question

Question: solve log₂(x − 1) + log₂(x + 1) = 3.

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 product rule, from the log rules lesson: plus outside the logs is times inside one log.

So we get one log: log₂((x − 1)(x + 1)).

Careful!

Plus between two logs is not plus inside.

It is times inside one log.

2 Write it as a power

A log is a power.

So 2 to the power 3 is what is inside.

So (x − 1)(x + 1) is 2³.

2³ is 8.

3 Open the brackets

Each part times each part, then like terms: x² − 1 = 8.

4 Get x² alone

There is no x term, so we get x² by itself.

To remove the − 1, we add 1 on both sides.

8 + 1 = 9.

Now we have x² = 9.

5 Square root of both sides

x squared is 9.

So x is the square root of 9, or minus it.

Minus times minus is plus, so the negative one works too.

3 times 3 is 9.

So x = 3 or x = −3.

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 = −3.

Inside a log we get −4.

There is no log of a negative number.

So x = −3 is a false answer.

Books call it extraneous.

Now x = 3.

The insides are 2 and 4: both positive.

Times them inside one log: log base 2 of 8 is 3.

It works!

Our answers were:

We cross out the false answer.

Only x = 3 is left.

Let's look back.

First: make one log.

Then: write it as a power.

Then: open the brackets.

Then: get x² alone.

Then: square root of both sides.

Last: check.

▶ Watch it solved