Prove that log_a(x) = (log_b(x)) / (log_b(a)), where a, b and x are positive and a ≠ 1, b ≠ 1.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. M1 for starting the proof by setting y = log_a(x) (or equivalent).
2. A1 for correctly converting the logarithmic form to exponential form: x = a^y.
3. M1 for taking the logarithm to base b of both sides: log_b(x) = log_b(a^y), and then correctly applying the power rule for logarithms: log_b(x) = y log_b(a).
4. A1 for correctly rearranging the equation to isolate y: y = log_b(x) / log_b(a), and concluding the proof by substituting y back: log_a(x) = log_b(x) / log_b(a).