Indices and Logarithms, Short Questions (Question 1 – 8)


Question 1
Solve the equation, log3 [log2(2x – 1)] = 2

Solution:
log3 [log2 (2x – 1)] = 2 ← (if log a N = x, N = ax)
log2 (2x – 1) = 32
log2 (2x – 1) = 9
2x – 1 = 29
x = 256.5



Question 2
Solve the equation,   l o g 16 [ l o g 2 ( 5 x   4 ) ] = l o g 9 3

Solution:
l o g 16 [ l o g 2 ( 5 x   4 ) ] = l o g 9 3 l o g 16 [ l o g 2 ( 5 x   4 ) ] = 1 4 log 9 3 = log 9 3 1 2 = 1 2 log 9 3 = 1 2 ( 1 log 3 9 ) = 1 2 ( 1 2 ) = 1 4 l o g 2 ( 5 x   4 ) = 16 1 4 l o g 2 ( 5 x   4 ) = 2 5 x   4 = 2 2 5 x = 8 x = 8 5



Question 3
Solve the equation, 5 log 4 x = 125

Solution:
5 log 4 x = 125 log 5 5 log 4 x = log 5 125 put log for both side ( log 4 x ) ( log 5 5 ) = 3 ( log 4 x ) ( 1 ) = 3 x = 4 3 = 64



Question 4
Solve the equation, 5 log 5 ( x + 1 ) = 9

Solution:
5 log 5 ( x + 1 ) = 9 log 5 5 log 5 ( x + 1 ) = log 5 9 log 5 ( x + 1 ) . log 5 5 = log 5 9 log 5 ( x + 1 ) = log 5 9 x + 1 = 9 x = 8



Question 5
Solve the equation, log 9 ( x 2 ) = log 3 2

Solution:
log 9 ( x 2 ) = log 3 2 log a b = log c b log c a log 3 ( x 2 ) log 3 9 = log 3 2 log 3 ( x 2 ) 2 = log 3 2 log 3 ( x 2 ) = 2 log 3 2 log 3 ( x 2 ) = log 3 2 2 x 2 = 4 x = 6



Question 6
Solve the equation, log 9 ( 2 x + 12 ) = log 3 ( x + 2 )

Solution:
log 9 ( 2 x + 12 ) = log 3 ( x + 2 ) log 3 ( 2 x + 12 ) log 3 9 = log 3 ( x + 2 ) log 3 ( 2 x + 12 ) = 2 log 3 ( x + 2 ) log 3 ( 2 x + 12 ) = log 3 ( x + 2 ) 2 2 x + 12 = x 2 + 4 x + 4 x 2 + 2 x 8 = 0 ( x + 4 ) ( x 2 ) = 0 x = 4  (not accepted) x = 2



Question 7
Solve the equation, log 4 x = 3 2 log 2 3

Solution:
log 4 x = 3 2 log 2 3 log 2 x log 2 4 = 3 2 log 2 3 log 2 x 2 = 3 2 log 2 3 log 2 x = 2 × 3 2 log 2 3 log 2 x = 3 log 2 3 log 2 x = log 2 3 3 x = 27



Question 8
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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