Question 441990
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There are three basic rules of logarithms:

#1. {{{log(B,(A)) + log(B,(C)) = log(B,(AC))}}}

#2. {{{log(B,(A)) - log(B,(C))= log(B,(A/C))}}}

#3. {{{C*log(B,(A)) = log(B,(A^C))}}}

In one place we will need this rule of exponents and roots

#4. {{{A^(1/2) = sqrt(A)}}} 

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{{{log(3,4) +log(3,2)+log(3,2)}}}

Use #1 on the first two terms:

{{{log(3,4*2)+log(3,2)}}}

{{{log(3,8)+log(3,2)}}}

Use #1 again

{{{log(3,8*2)}}}

{{{log(3,16)}}}

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{{{log(4)+3log(x) +log(y)}}}

Use #3 on the middle term

{{{log(4)+log(x^3) +log(y)}}}

Use #1 on the first two terms:

{{{log(4x^3) +log(y)}}}

Use #1 again

{{{log(4x^3y)}}}

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{{{log(3) +expr(1/2)log(x)-log(5)}}}

Use #3 on the middle term

{{{log(3) +log(x^(1/2))-log(5)}}}

Use #4

{{{log(3) +log(sqrt(x))-log(5)}}}

Use #1 on the first two terms:

{{{log(3sqrt(x))-log(5)}}}

Use #2

{{{log(3sqrt(x)/5)}}}

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{{{2ln(x)-ln(3)+ln(6)}}}

Use #3 on the first term:

{{{ln(x^2)-ln(3)+ln(6)}}}

Use #2 on the first two terms:

{{{ln(x^2/3)+ln(6)}}}

Use #1

{{{ln((x^2/3)(6))}}}

{{{ln(2x^2)}}}

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{{{3log(x)+log(4)-log(x)-log(6)}}}

Use #3 on the first term:

{{{log(x^3)+log(4)-log(x)-log(6)}}}

Factor -1 out of the last two terms:

{{{log(x^3)+log(4)-(log(x)+log(6))}}}

Use #1 on the first two terms:

{{{log(4x^3)-(log(x)+log(6))}}}

Use #1 on the two terms in the parentheses:

{{{log(4x^3)-(log(6x))}}}

{{{log(4x^3)-log(6x)}}}

Use #2

{{{log((4x^3)/(6x))}}}

{{{log((2x^2)/(3))}}}

Edwin</pre>