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Question
Express the following as a single logarithm:
`3"log"(5)/(8) + 2"log"(8)/(15) - (1)/(2)"log"(25)/(81) + 3`
Solution
`3"log"(5)/(8) + 2"log"(8)/(15) - (1)/(2)"log"(25)/(81) + 3`
= `3"log"(5)/(2^3) + 2"log"(2^3)/(3 xx 5) - (1)/(2)"log"(5^2)/(3^4) + 3"log"10`
= `3"log"5 - 3"log"2^3 + 2"log"2^3 - 2"log"3 - 2"log"5 - (1)/(2)"log"5^2 + (1)/(2)"log"3^4 + 3"log(2 xx 5)`
= `3"log"5 - 3 xx 3"log"2 + 2 xx 3"log"2 - 2"log"3 - 2"log"5 - (1)/(2) xx 2"log"5 + (1)/(2) xx 4"log"3 + 3"log"2 + 3"log5`
= 3log5 - 9log2 + 6log2 - 2log3 - 2log5 - log5 + 2log3 + 3log2 + 3log5
= 3log5
= log53
= log125.
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