Write as an expression of a single logarithm and simplify completlely

3Log(4)3+ Log(4)2-1/2log(4)36

Numbers in parthensis are subscripts

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To do this we need some basic principles in logarithm.

For positive a,A and B;

`log_aA+Log_aB = log_a(AB)`

`log_aA-Log_aB = log_a(A/B)`

`Plog_aA = log_a(A)^P`

`3Log(4)3+ Log(4)2-1/2log(4)36`

`= log_4(3^3)+log_4(2)-log_4(36)^(1/2)`

`= log_4(27)+log_4(2)-log_4(6)`

`= log_4[(27xx2)/6]`

`= log_4(9)`

*So the answer is ;*

`3Log(4)3+ Log(4)2-1/2log(4)36 = log_4(9)`

**Sources:**

`3log_4 3+log_4 2 -1/2 log_4 36=` `=2log_4 3+ log_4 3+log_4 2-1/2log_4(3^2 xx 2^2)=`

`=2log_4 3 +log_4 3xx2 -1/2( log_4 3^2 +log_4 2^2)=`

`=2log_4 3 +log_4 6-1/2(2log_4 3 +2log_4 2)=`

`=2log_4 3+log_4 6 -(log_4 3+ log_4 2)=`

`=2log_4 3 +log_4 6-log_4 3xx2 =`

`=log_4 3 +log_4 6- log_4 6=2log_4 3`

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