Solve the equation 3 × 2x+1 = 4 × 32x-3. Give your answer correct to 3 significant figures.

Mark scheme

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

1. [1] Use a correct law for the logarithm of a product or the logarithm of a power. Not available after incorrect manipulation, e.g. 6x+1 =122x-3.
2. [1] Obtain ln3+(x+1)ln2 = ln 4+ (2x-3) ln 3. Or equivalent with no powers.
3. [1] Solve for x. E.g. x = (ln(2/81)) / (ln(4/3)). Allow with logarithms evaluated.
4. [1] Obtain 2.46 only.
5. [1] Alternative Method: Use a correct law of indices for a product or a power. E.g. 2x+1 =2×2x. Not available after incorrect manipulation, e.g. 6x+1 =122x-3. 4 = 22 on its own is not enough.
6. [1] Alternative Method: Obtain (2/3)^x = 2/81. OE with powers of x only, e.g. 6×2x = 4/27 ×9x. Note: (3x)² is not far enough.
7. [1] Alternative Method: Solve for x. E.g. x = ln(2/81) / ln(2/3) or x = ln(1/81) / ln(2/3). Allow with logarithms evaluated.
8. [1] Alternative Method: Obtain 2.46 only.