Try to relax. Your desperation has surely progressed to the point where you're unable to think clearly, and to agonize over it any further would only cause you more pain and frustration. I've never seen this kind of problem before. But I arrived here in a calm state, having just finished my dinner and spent a few minutes rubbing my dogs, and I believe I've been able to crack the case.
Consider this: (2)^a negative power = (1/2)^the same power but positive.
So: Whatever power (2) must be raised to, in order to reach some number 'N', the same number 'N' can be reached by raising (1/2) to the same power but negative.
What I just said in that paragraph was: log₂ of(N) = <em>- </em>log(base 1/2) of (N) . I think that's the big breakthrough here. The rest is just turning the crank.
Now let's look at the problem:
log₂(x-1) + log(base 1/2) (x-2) = log₂(x)
Subtract log₂(x) from each side:
log₂(x-1) - log₂(x) + log(base 1/2) (x-2) = 0
Subtract log(base 1/2) (x-2) from each side:
log₂(x-1) - log₂(x) = - log(base 1/2) (x-2) Notice the negative on the right.
The left side is the same as log₂[ (x-1)/x ]
==> The right side is the same as +log₂(x-2)
Now you have: log₂[ (x-1)/x ] = +log₂(x-2)
And that ugly [ log to the base of 1/2 ] is gone.
Take the antilog of each side:
(x-1)/x = x-2
Multiply each side by 'x' : x - 1 = x² - 2x
Subtract (x-1) from each side:
x² - 2x - (x-1) = 0
x² - 3x + 1 = 0
Using the quadratic equation, the solutions to that are x = 2.618 and x = 0.382 .
I think you have to say that <em>x=2.618</em> is the solution to the original log problem, and 0.382 has to be discarded, because there's an (x-2) in the original problem, and (0.382 - 2) is negative, and there's no such thing as the log of a negative number.
