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Question
Find the value of the product: `|(log_3 64, log_4 3),(log_3 8, log_4 9)| xx |(log_2 3, log_8 3),(log_3 4, log_3 4)|`
Solution
`|(log_3 64, log_4 3),(log_3 8, log_4 9)| xx |(log_2 3, log_8 3),(log_3 4, log_3 4)| = |(log_3 64 * log_2 3 + log_4 3 * log_3 4, log_3 64 * log_8 3 + log_4 3 * log_3 4),(log_3 8 * log_2 3 + log_4 9 * log_3 4, log_2 8 * log_8 3 +log_4 9 * log_3 4)|`
= `|(log_2 64 + log_3 3, log_8 64 + log_3 3),(log_2 8 +log_3 9, log_8 8 + log_3 9)|`
`log_"b" "a" * log_"c" "b" = log_"c" "a"`
⇒ `log_"a" "a"` = 1
= `|(log_2 2^6 + 1, log_8 8^2 + 1),(log_2 2^3 + log_3 3^2, 1 + log_3 3^2)|`
= `|(6log_2 2 + 1, 2log_8 8 + 1),(3log_2 2 + 2log_3 3, 1 + 2log_3 3)|`
= `|(6 + 1, 2 + 1),(3 + 2, 1 + 2)|`
= `|(7, 3),(5, 3)|`
= 21 – 15
= 6
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