Logarithms

Traps in Logarithms

• Mixes up the laws of indices
  aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ. Check with small numbers: 2² × 2³ = 4 × 8 = 32 = 2⁵.
• Calculates a different quantity
  Name the quantity and its unit before you start (remainder; horizontal distance; cm²). After calculating, check your number is that quantity and not a neighbour.
• Misapplies logarithm laws
  log(ab) = log a + log b; log(a/b) = log a − log b. logₐ x = n ⇔ x = aⁿ (the base is raised to the answer).
• Inverts a relationship
  Say the rule with its direction (angle at centre = TWICE the angle at the circumference; (x − 1) = 0 gives x = +1). Check your answer against a simple case.
• Uses the wrong operation
  Write the rule in words first (bearing back = bearing ± 180°; IQR = Q₃ − Q₁), then substitute. Estimate: does the size of the answer make sense?
• Stops one step short of the answer
  Before writing the answer, re-read the question's last words (each angle? neither? Ama's share? variance?). Ask: is my number that thing, or a stage on the way to it?
• Multiplies logs that should be added
  log a + log b = log(ab): ADD the log values (3 + 2 = 5). Multiplying log values is never a log law.
• Incomplete inverse operations
  Undo operations in REVERSE order, one at a time, to the whole of each side. Make a list: what was done to x? Undo the last one first. Check by substituting back.
• Sign slip when moving or multiplying terms
  Moving a term across "=" means doing the opposite operation to BOTH sides. And (−)×(−) = (+). Write each step on a new line; circle every minus sign before you expand.