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solve the equation log3 (3x-6) = log3 (2x+1)

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3 Answers

Heather:
 
In this problem, you shouldn't be confused by the log3 since we know that when the log3 a = log3 b, then very simply a must be equal to b.  In this problem a=3x-6 and b=2x+1
 
So:  3x-6=2x+1
       3x-2x=1+6
            1x=7
             x=7
 
Hope this helps!
 
George T.
Hi Heather;
log3 (3x-6) = log3 (2x+1)
 
Before I answer this, I was like to briefly review logarithms.
Let's just say that...
log3 (3x-6) = 5
I randomly selected the number 5.
This would resolve as...
3x-6=35
Do you see how the base 3 moved to the other side of the = sign and became a 3, whereas the 5 rose to exponential status of 5?
I love the way Megan described the components as base, exponent and "answer".  If she does not mind, I intend to use it in future answers.
 
In the equation you provided, the base of 3 appears on both sides of the equation.  Henceforth...
log3 (3x-6) = log3 (2x+1)
we can cancel these.
3x-6=2x+1
Let's add 6 to both sides as we proceed to isolate x...
3x-6+6=2x+1+6
3x=2x+7
Let's subtract 2x from both sides...
-2x+3x=2x+7-2x
x=7
 
Let's verify...
(3x-6)  ???   (2x+1)
[3(7)-6] ??? [2(7)+1]
21-6  ???  14+1
15=15
All Good!
Hey Heather!
 
log39=2   -->example
3 is the base, 2 is the exponent and 9 is what I call the answer. You can rewrite this expression as 32=9 so you can see that. In the expression you have both of your logs have the same base of 3 and since they are set equal to each other tell you that they must have the same exponent. Therefore the answers must be equal so you can just set 3x-6=2x+1 and solve for x. 
In this similar problem:
log2(2x-2)=log2(x+1)
2x-2 = x+1    -->set answers equal to eachother
x-2=1            --> solve for x
x=3
 
Good Luck!