SOLUTION: what does condensing logarithm means

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Question 518340: what does condensing logarithm means
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
In regular algebra you have the commutative, associative,
and distributive properties.  In "logarithm algebra" you
have three more besides those.

1.   logb(A) + logb(C) = logb(AC)
2.   logb(A) + logb(C) = logb(A%2FC)
3.   C·logb(A) = logb(AC)


Condensing logarithms means using these 3 additional rules 
on them to write a simpler equivalent logarithm expression.
The "condensed" version is on the right of the above three 
equations, because what's on the right side is shorter and
more concise than the expression on the left.  Those same
rules can be used in reverse to "expand" the logarithm expression.

Example:

3%28log%285%2C%283x%29%29%2Blog%285%2C%28y%29%29%29-4log%285%2C%28%28x%29%29%29

Use rule 1 to change the sum of logs to the log of a producr:

3%28log%285%2C%283xy%29%29%29-4log%285%2C%28x%29%29

We don't need those big parentheses anymore:

3log%285%2C%283xy%29%29-4log%285%2C%28x%29%29


On both terms use rule 3 to change the coefficients to 
exponents of the quantity to which the log is being taken:

log%285%2C%28%283xy%29%5E3%29%29-log%285%2C%28x%5E4%29%29

Use a familiar rule of exponents to change %283xy%29%5E3 to 27x%5E3y%5E3,

log%285%2C%2827x%5E3y%5E3%29%29-log%285%2C%28x%5E4%29%29

Use rule 1 to change the sum of logs to the log of a producr:

log%285%2C%2827x%5E3y%5E3%2F%28x%5E4%29%29%29

Use a familiar rule of exponents to change %2827x%5E3y%5E3%29%2F%28x%5E4%29 to %2827y%5E3%29%2Fx

log%285%2C%28%2827y%5E3%29%2Fx%29%29

The condensed form consists of only one single logarithm.

Edwin