prove that log 81/8 - 2log3/2 + 3log2/3 + log3/4 = 0
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Step-by-step explanation:
log 81/8 - 2log 3/2 + 3log 2/3 + log 3/4 = 0
Consider LHS
= log 81/8 - 2log 3/2 + 3log 2/3 + log 3/4
Using Quotient rule log a/b = log a - log b
= log 81 - log 8 - 2( log 3 - log 2 ) + 3(log 2 - log 3 ) + log 3 - log 4
= log 81 - log 8 - 2log 3 + 2log 2 + 3log 2 - 3log 3 + log 3 - log 4
Using Power rule m.log a = log a^m
= log 81 - log 8 - log 3² + log 2² + log 2³ - log 3³ + log 3 - log 4
= log 81 - log 8 - log 9 + log 4 + log 8 - log 27 + log 3 - log 4
= log 81 - log 9 - log 27 + log 3
= log 81 + log 3 - ( log 9 + log 27 )
= log 243 - log 243
= 0
Hence proved.
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