MAT1510 ASSIGNMENT 3 2022

MAT1510
ASSIGNMENT 3
2022

,Solution:



1.1).


3
√(𝑥 3 + 1)3 (𝑥 2 + 5)4 √(𝑥 3 + 1)3 (𝑥 2 + 5)4
log 5 ( ) = 3 log 5 ( )
(𝑥 2 + 4𝑥 + 2)5 (𝑥 2 + 4𝑥 + 2)5

= 3 [log 5 (√(𝑥 3 + 1)3 (𝑥 2 + 5)4 ) − log 5 ((𝑥 2 + 4𝑥 + 2)5 )]
1
= 3 [log 5 ((𝑥 3 + 1)3 (𝑥 2 + 5)4 )2 − log 5((𝑥 2 + 4𝑥 + 2)5 )]

1
= 3 [ log 5 ((𝑥 3 + 1)3 (𝑥 2 + 5)4 ) − 5 log 5 (𝑥 2 + 4𝑥 + 2)]
2
3
= log 5 ((𝑥 3 + 1)3 (𝑥 2 + 5)4 ) − 15 log 5 (𝑥 2 + 4𝑥 + 2)
2




log 5 ((𝑥 3 + 1)3 (𝑥 2 + 5)4 ) = log 5 (𝑥 3 + 1)3 + log 5 (𝑥 2 + 5)4

= 3log 5 (𝑥 3 + 1) + 4 log 5 (𝑥 2 + 5)


3
√(𝑥 3 + 1)3 (𝑥 2 + 5)4 3
log 5 ( 2 5
) = log 5 ((𝑥 3 + 1)3 (𝑥 2 + 5)4 ) − 15 log 5 (𝑥 2 + 4𝑥 + 2)
(𝑥 + 4𝑥 + 2) 2

3
= [3log 5 (𝑥 3 + 1) + 4 log 5 (𝑥 2 + 5)] − 15 log 5 (𝑥 2 + 4𝑥 + 2)
2
9
= log 5 (𝑥 3 + 1) + 6log 5 (𝑥 2 + 5) − 15 log 5 (𝑥 2 + 4𝑥 + 2)
2

, 3
√(𝑥 3 + 1)3 (𝑥 2 + 5)4 9
log 5 ( ) = log 5 (𝑥 3 + 1) + 6log 5 (𝑥 2 + 5) − 15 log 5 (𝑥 2 + 4𝑥 + 2)
(𝑥 2 + 4𝑥 + 2)5 2



1.2).


16 16 4 1
log(𝑥 + 5) + log(𝑥 2 + 3) − log(𝑥 + 2) = 2 log(𝑥 + 5) + 4 log(𝑥 2 + 3) − log(𝑥 + 2)
8 4 8 2
1
= log(𝑥 + 5)2 + log(𝑥 2 + 3)4 − log(𝑥 + 2)2

= log(𝑥 + 5)2 + log(𝑥 2 + 3)4 − log √(𝑥 + 2)

= log[(𝑥 + 5)2 (𝑥 2 + 3)4 ] − log √(𝑥 + 2)

(𝑥 + 5)2 (𝑥 2 + 3)4
= log [ ]
√(𝑥 + 2)




Solution:



2.1).



𝑆𝑡𝑒𝑝1: 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 (𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑔(𝑥) 𝑏𝑦 2)