Indices and Logarithms, Short Questions (Question 9 – 14)


Question 9
Solve the equation,  log 2 4 x = 1 log 4 x

Solution:
log 2 4 x = 1 log 4 x log 2 4 x = 1 log 2 x log 2 4 log 2 4 x = 1 log 2 x 2 2 log 2 4 x = 2 log 2 x log 2 16 x 2 = log 2 4 log 2 x log 2 16 x 2 = log 2 4 x 16 x 2 = 4 x x 3 = 4 16 = 1 4 x = ( 1 4 ) 1 3 = 0.62996



Question 10
Solve the equation,  log 4 x = 25 log x 4

Solution:
log 4 x = 25 log x 4 1 log x 4 = 25 log x 4 1 25 = ( log x 4 ) 2 log x 4 = ± 1 5 log x 4 = 1 5        or       log x 4 = 1 5 4 = x 1 5                                 4 = x 1 5 x = 4 5                                  4 = 1 x 1 5 x = 1024                           x 1 5 = 1 4                                             x = 1 1024



Question 11
Solve the equation,  2 log x 5 + log 5 x = lg 1000

Solution:
2 log x 5 + log 5 x = lg 1000 2. 1 log 5 x + log 5 x = 3 × ( log 5 x )      2 + ( log 5 x ) 2 = 3 log 5 x ( log 5 x ) 2 3 log 5 x + 2 = 0 ( log 5 x 2 ) ( log 5 x 1 ) = 0 log 5 x = 2       or      log 5 x = 1 x = 5 2                               x = 5 x = 25



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
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)

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