NCEA Level 2LogarithmsAS 91261

Logarithms: 7 exam questions solved step by step

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

7 questions, one idea

  • 1. A log is an index: logₐ x = y, aʸ = x.
  • 2. Power, product and quotient laws.
  • 3. Unknown index: log both sides.
  • 4. Logs in it: combine, then index form.
Your turn

Question 1 · Achieved

  • (a) Write 6² = 36 in log form.
  • (b) Solve log₃ x = 4
Worked example

Q1: index form and log form

  • Base stays the base.
  • The log is the index.
  1. 6² = 36base 6, index 2
  2. log₆ 36 = 2the log is the index
  3. log₃ x = 4
  4. x = 3⁴index form
  5. = 3 × 3 × 3 × 3
  6. = 81
  7. 3⁴ = 81, so log₃ 81 = 4
Your turn

Question 2 · Achieved

  • Solve 4ˣ = 30
  • Give x to 3 decimal places.
Worked example

Q2: 4ˣ = 30

  • x is an index: use logs.
  • Round only at the end.
  1. 4ˣ = 30
  2. log 4ˣ = log 30log both sides
  3. x log 4 = log 30power law
  4. x = log 30 ÷ log 4÷ log 4
  5. = 2.453445...
  6. x = 2.453 (3 dp)
  7. 4^2.453 = 29.98 ≈ 30
  8. log(30 ÷ 4) = 0.875not log 30 ÷ log 4
Your turn

Question 3 · Merit

  • Write as a single log and evaluate:
  • 3 log 2 + log 5 − log 4
Worked example

Q3: 3 log 2 + log 5 − log 4

  • Powers first.
  • Then add, then subtract.
  1. 3 log 2 = log 2³power law
  2. = log 8
  3. log 8 + log 5 − log 4
  4. = log(8 × 5) − log 4product law
  5. = log 40 − log 4
  6. = log(40 ÷ 4)quotient law
  7. = log 10
  8. = 110¹ = 10
  9. 3 log 2 = log 6the 3 is a power
Your turn

Question 4 · Merit

  • Solve 5 × 2^(x + 1) = 120
  • Give x to 3 decimal places.
  • Careful: 5 × 2 is not the base.
Worked example

Q4: 5 × 2^(x + 1) = 120

  • Only the 2 is powered.
  • Divide by 5 first.
  1. 5 × 2^(x + 1) = 120
  2. 10^(x + 1) = 120only 2 is powered
  3. 2^(x + 1) = 24÷ 5 both sides
  4. (x + 1) log 2 = log 24log, power law
  5. x + 1 = log 24 ÷ log 2÷ log 2
  6. = 4.584962...
  7. x = 3.584962...− 1
  8. x = 3.585 (3 dp)
  9. 5 × 2^4.585 = 120.00
Your turn

Question 5 · Merit, in context

  • Earthquake magnitude: M = log(A ÷ A₀)
  • A: seismograph reading, A₀ fixed.
  • A magnitude 6.3 quake reads 2 mm.
  • Find the reading for a magnitude 7.8.
Worked example

Q5: magnitude 7.8 versus 6.3

  • Subtract: A₀ cancels.
  • Answer with units.
  1. 7.8 = log(A ÷ A₀)magnitude 7.8
  2. 6.3 = log(2 ÷ A₀)magnitude 6.3
  3. log(A ÷ A₀) − log(2 ÷ A₀) = 1.5
  4. log(A ÷ 2) = 1.5quotient law, A₀ cancels
  5. A ÷ 2 = 10^1.5index form, base 10
  6. A = 2 × 10^1.5× 2
  7. = 63.245... mm
  8. A = 63.2 mm (1 dp)
  9. 10^1.5 ≈ 31.6: about 32 times
Your turn

Question 6 · Excellence

  • Solve 3^(x + 1) = 5ˣ
  • Give x to 3 decimal places.
Worked example

Q6: 3^(x + 1) = 5ˣ

  • x on both sides.
  • Collect the x terms.
  1. (x + 1) log 3 = x log 5log, power law
  2. x log 3 + log 3 = x log 5expand
  3. log 3 = x log 5 − x log 3− x log 3
  4. log 3 = x(log 5 − log 3)factorise x
  5. x = log 3 ÷ (log 5 − log 3)÷ the bracket
  6. = log 3 ÷ log(5 ÷ 3)quotient law
  7. x = 2.150660... = 2.151 (3 dp)
  8. both sides = 31.86
Your turn

Question 7 · Excellence

  • Solve log₂(x + 2) + log₂(x − 4) = 4
Worked example

Q7: combine, then index form

  • One log first.
  • Then lose the log.
  1. log₂(x + 2) + log₂(x − 4) = 4
  2. log₂((x + 2)(x − 4)) = 4product law
  3. (x + 2)(x − 4) = 2⁴index form
  4. x² − 4x + 2x − 8 = 16expand, 2⁴ = 16
  5. x² − 2x − 8 = 16collect
  6. x² − 2x − 24 = 0− 16 both sides
Worked example

Q7: solve, reject, check

  • No log of a negative.
  • Check in the original.
  1. x² − 2x − 24 = 0
  2. (x − 6)(x + 4) = 0−6 × 4, −6 + 4
  3. x = 6 or x = −4
  4. x = −4log₂(−8) is undefined
  5. x = 6: log₂ 8 + log₂ 2
  6. = 3 + 1 = 4
  7. x = 6
Summary

The laws that solved all seven

  • A log is an index: switch forms, use the laws, check.
StepHow
Log formlogₐ x = y means aʸ = x
Powerlog aⁿ = n log a, do it first
Productlog a + log b = log(ab)
Quotientlog a − log b = log(a ÷ b)
aˣ = blog both sides, then divide
Logs in itcombine, then index form
Checkno log of a negative

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