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

Question

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

Answer 1

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.

Answer 2

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!

Answer 3

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! 

Answer 4

7

x=7

no matter what you have on two sides of an equation, however complicated, if all is the same except for one detail. set that one detail = from both sides. in this problem 3x-6 = 2x+1. 3x-2x=1+6 =7

Answer 5

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