5.6 Indices and Logarithms, SPM Practice (Long Questions)


5.6 Indices and Logarithms, SPM Practice (Long Questions)
Question 3:
Given that p= 3r and q = 3t, express the following in terms of rand/ or t.
(a)  (a)  log 3 p q 2 27 ,
(b)  log9p – log27 q.

Solution:
(a)
Given p = 3r, log3 p = r
q= 3t, log3 q =t

log 3 p q 2 27
= log3 pq2 – log327
= log3 p + log3 q2 – log3 33
= r + 2 log3 q – 3 log3 3
= r + 2 log3 q – 3(1)
= r + 2t – 3

(b)
log9 p– log27 q
= log 3 p log 3 9 log 3 q log 3 27 = r log 3 3 2 t log 3 3 3 = r 2 log 3 3 t 3 log 3 3 = r 2 t 3   



Question 4:
(a)  Simplify:
log2(2x + 1) – 5 log4 x2 + 4 log2 x
(b)  Hence, solve the equation:
log2(2x + 1) – 5 log4 x2 + 4 log2 x = 4

Solution:
(a)
log2 (2x + 1) – 5 log4 x2 + 4 log2 x
= log 2 ( 2 x + 1 ) 5 log 2 x 2 log 2 4 + 4 log 2 x = log 2 ( 2 x + 1 ) 5 2 log 2 x 2 + log 2 x 4 = log 2 ( 2 x + 1 ) log 2 ( x 2 ) ( 5 2 ) + log 2 x 4
= log 2 ( 2 x + 1 ) log 2 x 5 + log 2 x 4 = log 2 ( 2 x + 1 ) ( x 4 ) x 5 = log 2 2 x + 1 x

(b)
log2 (2x + 1) – 5 log4 x2 + 4 log2 x = 4
log 2 2 x + 1 x = 4        2 x + 1 x = 2 4        2 x + 1 x = 16        2 x + 1 = 16 x          14 x = 1              x = 1 14


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