Hint
Every original logarithm argument is strictly positive. Combine the logs, then enforce both argument restrictions.
Worked solution
(a)
\(\left(x - 10\right) \left(x - 9\right) = 12\) gives \(\displaystyle x=13\) or 6. The original arguments require x>10, so only 13 is valid.
Answer: \(13\) with \(x > 10\)
(b)
Check the original arguments.
\(6\) makes x−h negative.
Answer: \(6\)
(c)
Substitute the valid x-value.
\(3\)
Answer: \(3\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 3 marks
- 1 mark: State the positivity conditions of both logarithm arguments.
- 1 mark: Combine logarithms, solve the resulting quadratic and reject the inadmissible root.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Exponentiating does not make a negative logarithm argument valid.
(b) · 2 marks
- 1 mark: Check the original arguments.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Check the original domain, not just the combined argument.
(c) · 2 marks
- 1 mark: Substitute the valid x-value.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Exponentiating does not make a negative logarithm argument valid.
Total: 7 marks.