Solve the equation 2^(3x-4) = 3/5^x. Give your answer in the form (ln m)/(ln n), where m and n are integers.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.

1. Use log quotient law, e.g. ln (3/5^x) = ln 3 – ln 5^x or log product law, e.g. ln5^x + ln(2^(3x-4))
2. Use log power law, e.g. ln 5^x = xln5 or ln (2^(3x-4)) = (3x – 4) ln 2
3. Obtain correct expression for x in any exact form, e.g. (ln 3 + 4 ln 2) / (3 ln 2 + ln 5)
4. Obtain final answer ln 48 / ln 40
5. Rearrange to obtain (2³ × 5)^x = 3 × 2⁴
6. Use log power law on an equation of the form a^(bx) = c e.g. x ln (2³ × 5) = ln (3 × 2⁴)
7. Obtain correct expression for x in any exact form, e.g. x = ln(3×2⁴) / ln(2³×5)
8. Obtain final answer ln 48 / ln 40