SOLUTION: {{{log(3,3x+6)-log(3,x-6)=2}}}

Question 34436:
Found 3 solutions by mukhopadhyay, dimndskier, Prithwis:
Answer by mukhopadhyay(490)   (Show Source): You can put this solution on YOUR website!
log(3,3x+6)-log(3,x-6)=2
=> log[(3x+6)/(x-6)] = 2
=> [(3x+6)/(x-6)] = 10^2 = 100
=> (3x+6) = 100(x-6)
=> 3x+6 = 100x-600
=> 97x = 606
=> x = 606/97
Answer by dimndskier(8)   (Show Source): You can put this solution on YOUR website!
First you have to understand that the subtraction (to keep wording and explanation simple) of two logarithmic expressions is equal to the logarithm of them in a fraction, no matter what the base is.

Therefore if I use 9 and substitute in for x, we get:

Before I help with this problem, you must understand what the purpose of logarithms really are.
To put it simply, sometimes you know the BASE value of an expression (BASE), and in the same expression you don't know the power (EXPONENT) that produces the answer (ANSWER).

So, here's an easy way to remember WHY you use this mathematical function:

How does this work?
Well, we know that , so , get it?
In your problem however there is the necessity of keep tracking of the algebraic expressions as the functions of the logarithms.

Here is how we work this problem, really...
, this can be rewritten as:

Don't forget why we use logarithm... to find the answer when we know the base and answer.
So, we have the base of 3 and exponent of 2, which means and then is simplified to 9!
Now we REALLY have to solve this problem:

, which then becomes...
==>
, which simplifies to:
, and then we discover that x=10
The final answer is x=6.

Answer by Prithwis(166)   (Show Source): You can put this solution on YOUR website!
log(3,3x+6)-log(3,x-6)=2
=> log[(3x+6)/(x-6)] = 2
=> [(3x+6)/(x-6)] = 10^2 = 100
=> (3x+6) = 100(x-6)
=> 3x+6 = 100x-600
=> 97x = 606
=> x = 606/97

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