Solve 3 × 2^(x - 1) = 12 Step by Step

Step by step

The question

Question: solve 3 times 2 to the power (x − 1) = 12.

x is up in the power.

3 × 2^{x − 1} = 12

Careful!

The power is only on the 2.

Powers come before times.

So we can't join 3 and 2 first.

Careful: not 3 × 2^{(x − 1)} = 6^{(x − 1)} ✗

1 Get the power alone

First the power must be alone: nothing added to it, nothing times it.

first: the power alone

The power is times 3.

To remove times 3, we divide by 3.

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.

3 × 2^{x − 1}3 = 123

3 divided by 3 is 1.

So just the power is left.

12 divided by 3 is 4.

Now we have 2 to the power (x − 1) = 4.

2^{x − 1} = 4

2 Write 4 as a power of 2

4 is 2 times itself, 2 times.

2 × 2 = 4

So 4 is 2².

Now both sides are 2 to a power.

2^{x − 1} = 2²

3 The powers are equal

Both sides are 2 to a power.

Same base, and they are equal.

So the powers must be equal too.

same base → same power

Now we have x − 1 = 2.

x − 1 = 2

4 Remove the − 1

We want x by itself.

The − 1 is in the way.

To remove − 1, we add 1.

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 ~~− 1 + 1~~ = 2 + 1

− 1 and + 1 make zero.

Cross them out.

2 + 1 = 3.

2 + 1 = 3

So x = 3!

x = 3

5 Check

Let's check in the question: put 3 in place of x.

Check: 3 × 2^{3 − 1} = 3 × 4 = 12

We get 12, the same as the right side.

So x = 3 is right!

Let's look back.

First: get the power alone.

Then: write 4 as a power of 2.

Then: the powers are equal.

Then: remove the − 1.

Last: check.

Answer