NCEA Level 2LogarithmsAS 91261

Logarithms: what they mean and the log laws

What this lesson covers

  • A logarithm is the index: log base a of x is the power a is raised to to give x.
  • The log laws are the index laws written in log form.
Key idea

A log answers one question

  • log₂ 8 asks: 2 to what power is 8?
  • The answer is 3, because 2³ = 8.
  • A logarithm is an index.
Worked example

Two ways to write one fact

  • Index form: 2³ = 8.
  • Log form: log₂ 8 = 3.
  • Same base, and the power becomes the answer.
Two ways to write one fact
Worked example

Reading logs as questions

  • Ask the question the log is asking.
  • log₅ 1 = 0, because 5⁰ = 1.
  • log₃(1/9) = −2, because 3⁻² = 1/9.
LogAsksAnswer
log₂ 82 to what is 8?3
log₁₀ 100010 to what is 1000?3
log₅ 15 to what is 1?0
log₃(1/9)3 to what is 1/9?−2
Key idea

The log laws are the index laws

  • Multiplying adds indices, so logs add.
  • Dividing subtracts them.
  • A power inside comes out the front.
Log ofEqualsBecause
log(ab)log a + log bmultiply: add indices
log(a/b)log a − log bdivide: subtract
log(aⁿ)n log apower: multiply
Worked example

Combining logs

  • Powers first.
  • Then add or subtract.
  1. 2 log 3 + log 5
  2. = log 3² + log 5power law
  3. = log 9 + log 5
  4. = log(9 × 5)add: multiply inside
  5. = log 45
Worked example

Solving 3ˣ = 20

  • The unknown is an index.
  • Take logs of both sides.
  1. 3ˣ = 20
  2. log 3ˣ = log 20log both sides
  3. x log 3 = log 20power law
  4. x = log 20 ÷ log 3÷ log 3
  5. = 2.727
  6. 3^2.727 ≈ 20.00check
Worked example

A harder index: 5^(2x − 1) = 80

  • Brackets round the index.
  • Keep 4 decimal places.
  1. 5^(2x − 1) = 80
  2. (2x − 1) log 5 = log 80logs, power law
  3. 2x − 1 = log 80 ÷ log 5÷ log 5
  4. = 2.7227
  5. 2x = 3.7227add 1
  6. x = 3.7227 ÷ 2 = 1.861÷ 2
Worked example

An equation with logs in it

  • Combine into one log.
  • Switch to index form.
  1. log₂ x + log₂(x + 2) = 3
  2. log₂(x(x + 2)) = 3add: multiply inside
  3. x(x + 2) = 2³index form
  4. x² + 2x = 8expand
  5. x² + 2x − 8 = 0subtract 8
  6. (x + 4)(x − 2) = 0factorise
  7. x = 2 or x = −4
Key idea

Check in the original equation

  • Substitute back.
  • No logs of negatives.
  1. x = 2 or x = −4
  2. x = 2: log₂ 2 + log₂ 4
  3. = 1 + 2 = 3
  4. x = −4: log₂(−4)no log of a negative
  5. x = 2
Key idea

Where the marks are

  • Achieved: solve simple log and index equations.
  • Merit: manipulate with the log laws.
  • Excellence: logs inside a real model.
Summary

What to take away

  • A log is an index. The log laws are the index laws.
logₐ x = ymeans aʸ = x
Multiply insideadd the logs
Divide insidesubtract the logs
Power insidebring it to the front
aˣ = bx = log b ÷ log a
Log equationscombine, then index form
Your turn

Your turn: four questions

  • Pause here and answer each one.
  • Question 1: ask the question the log asks.
  • Question 4: switch to index form.
Question
1Evaluate log₄ 64.
2Write 2 log 3 + log 5 as a single log.
3Solve 2ˣ = 50, to 3 decimal places.
4Solve log₃(x + 1) = 2.
Show answer

1. 4³ = 64, so 3. 2. log 9 + log 5 = log 45. 3. x = log 50 ÷ log 2 = 5.644. 4. x + 1 = 3² = 9, so x = 8.

Worked example

The answers: questions 1 and 2

  1. 1. 4 to what power is 64?
  2. 4³ = 64, so log₄ 64 = 3
  3. 2. 2 log 3 = log 3² = log 9power law
  4. log 9 + log 5 = log(9 × 5)add: multiply inside
  5. = log 45
Worked example

The answers: questions 3 and 4

  1. 3. 2ˣ = 50
  2. x log 2 = log 50logs, power law
  3. x = log 50 ÷ log 2÷ log 2
  4. = 5.644
  5. 4. log₃(x + 1) = 2
  6. x + 1 = 3²index form
  7. x + 1 = 9, so x = 8
  8. log₃ 9 = 2check

Is your child on track?

Take the free placement test