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Exponential equations and logarithms

Algebra 2 · 22 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Write it as a power
  2. Work out the power
  3. Remove the + 1
  4. Remove the 2
  5. Check

The steps for Solve log_3(2x + 1) = 2. Every solution names its steps, so you always know what comes next.

Worked examples

Solve 2^x = 16 Easy

The question

Question: solve 2 to the power x = 16.

x is up in the power.

2x = 16

1 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 = 24

2 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 = 4!

x = 4

3 Check

Let's check: put 4 in place of x.

Check: 24 = 16

We get 16, the same as the right side.

So x = 4 is right!

Let's look back.

First: write 16 as a power of 2.

Then: the powers are equal.

Last: check.

Answer
x = 4

▶ Watch it solved

Solve 4^x = 8 Medium

The question

Question: solve 4 to the power x = 8.

x is up in the power.

4x = 8

1 Write both as powers of 2

4 is 2², and 8 is 2³.

We write both as powers of 2.

2 × 2 = 4
2 × 2 × 2 = 8

So 4 becomes 2², and 8 becomes 2³.

(22)x = 23

A power of a power: multiply the powers.

(am)n = am × n

The powers multiply: 2 times x is 2x.

22x = 23

2 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 2x = 3.

2x = 3

3 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.

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.

2x2 = 32

2 divided by 2 is 1.

So just x is left.

So x = 3/2!

x = 32

4 Check

Let's check: put 3/2 in place of x.

Check: 432 = (22)32 = 23 = 8

We get 8, the same as the right side.

So x = 3/2 is right!

Let's look back.

First: write both as powers of 2.

Then: the powers are equal.

Then: remove the 2.

Last: check.

Answer
x = 32

▶ Watch it solved

Solve log_3(2x + 1) = 2 Harder

The question

Question: solve log₃(2x + 1) = 2.

log3(2x + 1) = 2

1 Write it as a power

A log is a power.

log₃(2x + 1) = 2 means: 3 to the power 2 is 2x + 1.

log3 c = k means 3k = c

So 2x + 1 is 3².

2x + 1 = 32

2 Work out the power

3 times 3 is 9.

3 × 3 = 9

Now we have 2x + 1 = 9.

2x + 1 = 9

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 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.

2x + 1 − 1 = 9 − 1

+ 1 and − 1 make zero.

Cross them out.

9 − 1 = 8.

9 − 1 = 8

Now we have 2x = 8.

2x = 8

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.

2x2 = 82

2 divided by 2 is 1.

So just x is left.

8 divided by 2 is 4.

82 = 4

So x = 4!

x = 4

5 Check

Let's check: put 4 in place of x.

Check: log3(2(4) + 1) = log3(8 + 1) = log3 9

log₃ 9 is 2, because 3² is 9.

So x = 4 is right!

log3 9 = 2, because 32 = 9

Let's look back.

First: write it as a power.

Then: work out the power.

Then: remove the + 1.

Then: remove the 2.

Last: check.

Answer
x = 4

▶ Watch it solved

Practice

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

Easy

  1. 2^x = 16
    Answer
    x = 4
  2. 3^x = 81
    Answer
    x = 4
  3. 2^(x+1) = 32
    Answer
    x = 4
  4. 5^(2x - 1) = 125
    Answer
    x = 2
  5. log_2(8)
    Answer
    log2 8 = 3
  6. log_3(9)
    Answer
    log3 9 = 2
  7. log(1000)
    Answer
    log 1000 = 3
  8. log_5(25)
    Answer
    log5 25 = 2

Medium

  1. 4^x = 8
    Answer
    x = 32
  2. 9^x = 27
    Answer
    x = 32
  3. 3^x = 20
    Answer
    x ≈ 2.727
  4. log_2(1/8)
    Answer
    log2(18) = −3
  5. log_2(4 × 8)
    Answer
    log2(4 × 8) = 5
  6. log_3(81 ÷ 3)
    Answer
    log3(813) = 3
  7. log_2(8^2)
    Answer
    log2(82) = 6
  8. log_2(x) = 5
    Answer
    x = 32

Harder

  1. log_3(2x + 1) = 2
    Answer
    x = 4
  2. 27^x = 9
    Answer
    x = 23
  3. log_5(1/25)
    Answer
    log5(125) = −2
  4. log_4(x - 1) = 2
    Answer
    x = 17
  5. 2^(3x) = 4^(x+1)
    Answer
    x = 2
  6. log(0.01)
    Answer
    log 0.01 = −2