Question

Simplify the following:

`3"log" (32)/(27) + 5 "log"(125)/(24) - 3"log" (625)/(243) + "log" (2)/(75)`

Answer 1

`3"log" (32)/(27) + 5 "log"(125)/(24) - 3"log" (625)/(243) + "log" (2)/(75)`

= `3"log" (2^5)/(3^3) + 5"log"(5^3)/(2^3 xx 3) - 3"log"(5^4)/(2 xx 3^4) + "log"(2)/(3 xx 5^2)`

= 3 log 2⁵ − 3 log 3³ + 5 log 5³ − 5 log 2³ − 5 log 3 − 3 log 5⁴ + 3 log 2 + 3 log 3⁴ + log 2 − log 3 - log 5²

= 3 x 5 log 2 − 3 x 3 log 3 + 5 x 3 log 5 − 5 x 3 log 2 − 5 log 3 − 3 x 4 log 5 + 3 log 2 + 3 x 4 log 3 + log 2 − log 3 − 2 log 5

= 15 log 2 − 9 log 3 + 15 log 5 − 15 log 2 − 5 log 3 − 12 log 5 + 3 log 2 + 12 log 3 + log 2 − log 3 − 2 log 5

= log 5 + log 2

= log (5 x 2)

= log 10

= 1.