Thread: Another Logarithm Equation: Not sure how to solve for x properly.

Hi again,

I am stuck on another logarithmic equation:

log3 x - 2 = log3 (x - 8)

Now, I assume log3 cancels out, which leaves:

x - 2 = x - 8

But from there, I am unsure how you would get x by itself, because if you try and transfer x to just one side, it cancels out because you have x and -x. I think I am on the wrong track though, so all guidance will be much appreciated once again.

Thanks,

Nathaniel

Originally Posted by BinaryBoy

Hi again,

I am stuck on another logarithmic equation:

log3 x - 2 = log3 (x - 8)

Now, I assume log3 cancels out, which leaves:

x - 2 = x - 8 Mr F says: It should be clear that this is nonsense as it implies -2 = -8.

But from there, I am unsure how you would get x by itself, because if you try and transfer x to just one side, it cancels out because you have x and -x. I think I am on the wrong track though, so all guidance will be much appreciated once again.

Thanks,

Nathaniel

The equation is possibly meant to be (and your job is to fill in the missing steps in getting this).

Now solve for x.

If the problem was, indeed, , then what you did is exactly correct- ln(x) is a "one-to-one function" so it must be true that x- 2= x- 8. But no matter what x is, that is the same as -2= -8 which is not true. NO value of x makes that equation true.

But,as mr fantastic said, your lack of parentheses makes this problem ambiguous. It might be , , , or . The first and last of those, which would be the most reasonable interpretations, have no solution!

Thanks for the help Mr Fantastic and HallsofIvy. I have solved for x and checked my answer and the equation is indeed of the form: log3 (x) -2 = logx (x - 8).

So, what I did was:

log3(x) = logx (x - 8) + 2

Then I moved the logx(x - 8) to the other side by dividing both sides. So I get:

log3(x / x - 8) = 2

This turns into:

log3( x / x - 8) = (3 to the power of 2)
= 9

Then I multiplied the x - 8 back over:

log3(x) = 9(x - 8)

log3(x) = 9x - 72

Multiply the x over as well:

0 = 8x - 72

Move the 72 to the other side

8x = 72

Divide by 8 to yield:

x = 9.

Cool! Many thanks for the help!

Nathaniel

You should, of course, check that answer in the original problem:


Setting x= 9, and so the equation becomes 2- 2= 0 which is, in fact, correct.