Solve the equation 9^(5x) / 27^(x-2) = 243.

Mark scheme

Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.

1. [1] (a) 3^(10x) / 3^(3x-6) [= 243] oe or log 9^(5x)-log 27^(x-2) = log 243 oe
2. [1] (a) 3^(7x+6) = 3^5 soi oe or 5x(log9)-(x-2) log 27 = log 243
3. [1] (a) x=-1/7
4. [2] (b) 1/2 log_a b - 1/2 = 1/log_a b or 1/2 log_a b - 1/2 = log_a a / log_a b
5. [1] (b) Clears the fraction and rearranges (log_a b)² - 1/2 log_a b = 1 oe or (log_a b)² - log_a b - 2 = 0 oe or let x = log_a b => x² - x - 2 = 0 oe or 1/2 - 1/2 log_a a = (log_a a)² or 0 = 2(log_a a)² + log_a a -1 oe or let y = log_b a => 2y² + y - 1 = 0
6. [1] (b) (log_a b-2)(log_a b+1) oe or (2log_a a-1)(log_a a+1)
7. [1] (b) [log_a b=2, log_a b=-1 or log_b a = 1/2, log_b a = -1 leading to ] b = a², b = a^-1