Theory question 110
Indices and Logarithms(a) Solve: (1/4)(m + 1)/3 + 4 = 5(1/2)(m + 1)/3
(b) Given that p log3 x = log81 x, find the value of p
Show the model answer
Model answer
(a) Let u = (½)^((m + 1)/3); then (¼)^((m + 1)/3) = u².
u² − 5u + 4 = 0, (u − 1)(u − 4) = 0, u = 1 or 4.
u = 1: (m + 1)/3 = 0, m = −1.
u = 4 = (½)⁻²: (m + 1)/3 = −2, m = −7.
m = −1 or m = −7.
(b) log₈₁ x = log₃ x / log₃ 81 = ¼ log₃ x, so p = ¼.
Explanation
A substitution turns the index equation into a quadratic.