3 (a) Write 1+1g(x² - 1) - 21g(x-1), where x > 1, as a single logarithm to base 10. Give your answer in its simplest form. (b) Solve the equation 4 log5(x+1) = 9log(x+1) 5, giving your answers in the form a+b√c, where a, b and c are constants.

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

1. [4] B1 (for 3a) 1=lg10 soi
2. [4] M1 For correct use of logs power rule or multiplication or division rule. i.e.: Award M1 for example: 2log(x-1) = log(x – 1)² or log(x² - 1) = log(x - 1) + log(x + 1) or log10 + log(x² - 1) = log10(x² -1) or log(x² - 1) - log(x – 1)² = log (x²-1)/(x-1)²
3. [4] DM1 Dep on previous M1 for a correct attempt to factorise and an attempt to simplify
4. [4] A1 (lg(10(x+1))/(x-1))
5. [5] B1 For change of base
6. [5] M1 for a correct method in forming a quadratic equation Dep on previous B1
7. [5] M1 Dep on dealing with logarithms correctly Allow if ± is missing
8. [5] A1 x=-1+5√5, −1-√5/4