O lvl A Maths: Indices and Logarithms

Question from http://www.sgforums.com/forums/2297/topics/322475

1) If a^2x-1 = b^1-3y and a^3x-1 = b^2y-2, show that 13xy = 7x + 5y - 3.

2) Given that log₅ x = 4 log_x 5, calculate the possible values of x.

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Answer:

1)

Log to base 10 on the equations

lg (a^2x-1) = lg (b^1-3y)
(2x - 1) lg a = (1 - 3y) lg b
(2x - 1) /(1 - 3y) = lg b / lg a----(1)

lg (a^3x-1) = lg (b^2y-2)
(3x - 1) lg a = (2y - 2) lg b
(3x-1) /(2y-2) = lg b / lg a----(2)

(1) = (2)

thus
(2x - 1) /(1 - 3y) = (3x-1) /(2y-2)
4xy + 2 - 4x- 2y = 3x + 3y- 1- 9xy

Hence 13xy = 5y + 7x - 3

2) log₅ x = 4 log_x 5
log₅ x = 4 log₅ 5 / log₅ x

[log₅ x]² = 4

log₅ x = ±2

x=5^-2 or 5²

x= ¹/₂₅ or 25

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