5.5.3 Indices and Logarithms, Short Questions (Question 11 – 13)


Question 11
Given that 2 log2 (xy) = 3 + log2x + log2 y
Prove that x2 + y2– 10xy = 0.

Solution:
2 log2 (xy) = 3 + log2x + log2 y
log2 (xy)2 = log2 8 + log2 x + log2y
log2 (xy)2 = log2 8xy
(xy)2 = 8xy
x2– 2xy + y2 = 8xy
x2 + y2 – 10xy = 0 (proven)


Question 12
Solve the equation,  log 2 5 x + log 4 16 x = 6

Solution:
log 2 5 x + log 4 16 x = 6 log 2 5 x + log 2 16 x log 2 4 = 6 log 2 5 x + log 2 16 x 2 = 6 2 log 2 5 x + log 2 16 x = 12 log 2 ( 5 x ) 2 + log 2 16 x = 12 log 2 ( 25 x ) + log 2 16 x = 12 log 2 ( 25 x ) ( 16 x ) = 12 log 2 400 x 2 = 12 400 x 2 = 2 12 x 2 = 10.24 x = 3.2



Question 13
Solve the equation, 2 log 5 2 = log 2 ( 2 x )

Solution:
2 log 5 2 = log 2 ( 2 x ) 2 = log 5 2. log 2 ( 2 x ) 2 = 1 log 2 5 . log 2 ( 2 x ) 2 log 2 5 = log 2 ( 2 x ) log 2 5 2 = log 2 ( 2 x ) 25 = 2 x x = 23

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