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Exponential equations

Power alone first · Algebra 2 · 22 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Get the power alone
  2. Write 16 as a power of 2
  3. The powers are equal
  4. Remove the + 1
  5. Check

The steps for Solve 3 × 2^(x + 1) = 48. Every solution names its steps, so you always know what comes next.

Worked examples

Solve 3 × 2^x = 24 Easy

The question

Question: solve 3 times 2 to the power x = 24.

x is up in the power.

3 × 2x = 24

Careful!

The power is only on the 2.

Powers come before times.

So we can't join 3 and 2 first.

Careful: not 3 × 2x = 6x ✗

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 × 2x3 = 243

3 divided by 3 is 1.

So just the power is left.

24 divided by 3 is 8.

Now we have 2ˣ = 8.

2x = 8

2 Write 8 as a power of 2

8 is 2 times itself, 3 times.

2 × 2 × 2 = 8

So 8 is 2³.

Now both sides are 2 to a power.

2x = 23

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

So x = 3!

x = 3

4 Check

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

Check: 3 × 23 = 3 × 8 = 24

We get 24, the same as the right side.

So x = 3 is right!

Let's look back.

First: get the power alone.

Then: write 8 as a power of 2.

Then: the powers are equal.

Last: check.

Answer
x = 3

▶ Watch it solved

Solve 2 × 3^x - 4 = 50 Medium

The question

Question: solve 2 times 3 to the power x, − 4 = 50.

x is up in the power.

2 × 3x − 4 = 50

Careful!

The power is only on the 3.

Powers come before times.

So we can't join 2 and 3 first.

Careful: not 2 × 3x = 6x ✗

1 Get the power alone

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

first: the power alone

We want the power by itself.

The − 4 is in the way.

To remove − 4, we add 4.

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.

2 × 3x − 4 + 4 = 50 + 4

− 4 and + 4 make zero.

Cross them out.

50 + 4 is 54.

Now we have 2 × 3ˣ = 54.

2 × 3x = 54

The power is times 2.

To remove times 2, we divide by 2.

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

2 × 3x2 = 542

2 divided by 2 is 1.

So just the power is left.

54 divided by 2 is 27.

Now we have 3ˣ = 27.

3x = 27

2 Write 27 as a power of 3

27 is 3 times itself, 3 times.

3 × 3 × 3 = 27

So 27 is 3³.

Now both sides are 3 to a power.

3x = 33

3 The powers are equal

Both sides are 3 to a power.

Same base, and they are equal.

So the powers must be equal too.

same base → same power

So x = 3!

x = 3

4 Check

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

Check: 2 × 33 − 4 = 2 × 27 − 4 = 54 − 4
= 50

We get 50, the same as the right side.

So x = 3 is right!

Let's look back.

First: get the power alone.

Then: write 27 as a power of 3.

Then: the powers are equal.

Last: check.

Answer
x = 3

▶ Watch it solved

Solve 3 × 2^(x + 1) = 48 Harder

The question

Question: solve 3 times 2 to the power (x + 1) = 48.

x is up in the power.

3 × 2x + 1 = 48

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 × 2x + 13 = 483

3 divided by 3 is 1.

So just the power is left.

48 divided by 3 is 16.

Now we have 2 to the power (x + 1) = 16.

2x + 1 = 16

2 Write 16 as a power of 2

16 is 2 times itself, 4 times.

2 × 2 × 2 × 2 = 16

So 16 is 2⁴.

Now both sides are 2 to a power.

2x + 1 = 24

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 = 4.

x + 1 = 4

4 Remove the + 1

We want x by itself.

The + 1 is in the way.

To remove + 1, we take away 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 = 4 − 1

+ 1 and − 1 make zero.

Cross them out.

4 − 1 = 3.

4 − 1 = 3

So x = 3!

x = 3

5 Check

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

Check: 3 × 23 + 1 = 3 × 16 = 48

We get 48, the same as the right side.

So x = 3 is right!

Let's look back.

First: get the power alone.

Then: write 16 as a power of 2.

Then: the powers are equal.

Then: remove the + 1.

Last: check.

Answer
x = 3

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. 3 × 2^x = 24
    Answer
    x = 3
  2. 2 × 3^x = 18
    Answer
    x = 2
  3. 5 × 2^x = 40
    Answer
    x = 3
  4. 4 × 3^x = 36
    Answer
    x = 2
  5. 2^x + 5 = 37
    Answer
    x = 5
  6. 3^x + 2 = 29
    Answer
    x = 3
  7. 2^x - 3 = 13
    Answer
    x = 4
  8. 5^x - 4 = 21
    Answer
    x = 2

Medium

  1. 2 × 3^x - 4 = 50
    Answer
    x = 3
  2. 3 × 2^x + 1 = 25
    Answer
    x = 3
  3. 4 × 2^x - 8 = 24
    Answer
    x = 3
  4. 2 × 5^x + 3 = 53
    Answer
    x = 2
  5. 5 × 2^x = 60
    Answer
    x ≈ 3.585
  6. 3 × 2^x = 45
    Answer
    x ≈ 3.907
  7. 2e^x = 14
    Answer
    x ≈ 1.946
  8. 3e^x = 21
    Answer
    x ≈ 1.946
  9. e^x + 4 = 10
    Answer
    x ≈ 1.792

Harder

  1. 3 × 2^(x + 1) = 48
    Answer
    x = 3
  2. 2 × 3^(2x) = 162
    Answer
    x = 2
  3. 3 × 2^(x - 1) = 12
    Answer
    x = 3
  4. 2 × 5^x - 1 = 9
    Answer
    x = 1
  5. 2^x + 9 = 1
    Answer
    No power of 2 is −8.