Hint
Both logarithm bases are positive and different from one. Take base-2 logarithms of the argument and divide by that of the base.
Worked solution
(a)
Take base-2 logarithms of the argument and divide by that of the base.
\(\displaystyle \frac{5}{2}\)
Answer: \(\displaystyle \frac{5}{2}\)
(b)
Take base-2 logarithms of the argument and divide by that of the base.
\(\displaystyle \frac{2}{5}\)
Answer: \(\displaystyle \frac{2}{5}\)
(c)
Take base-2 logarithms of the argument and divide by that of the base.
\(\displaystyle \frac{5}{4}\)
Answer: \(\displaystyle \frac{5}{4}\)
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) · 2 marks
- 1 mark: Take base-2 logarithms of the argument and divide by that of the base.
- 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: The denominator is the logarithm of the original base.
(b) · 2 marks
- 1 mark: Take base-2 logarithms of the argument and divide by that of the base.
- 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: Do not invert only one part of a change-of-base expression.
(c) · 2 marks
- 1 mark: Take base-2 logarithms of the argument and divide by that of the base.
- 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: The denominator is the logarithm of the original base.
Total: 6 marks.