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Ok, I am fairly new to working with logarithms, but things had been going fairly well up until now. I had some minor mistakes here and there, and had gotten over the initial "hump" when you start something new, but now I am completely locked. The problem is:
(3^2x - 6 * 3^x) / (2 * 3^x + 3) = -1
So, I add 1 to both sides, and then multiply it by (2 * 3^x + 3) / (2 * 3^x + 3) to get:
(3^2x - 6 * 3^x) / (2 * 3^x + 3) + (2 * 3^x + 3) / (2 * 3^x + 3) = 0
I then add them together, and get:
(3^2x - 4 * 3^x + 3) / (2 * 3^x + 3) = 0
From here, I know that I can safely forget about the denominator, as it is only the numerator that is going to give me 0, so now I am at:
(3^2x - 4 * 3^x + 3) = 0
I put that through the quadratic formula, and find that 3^x is equal to 4, or 0. So I change this to a logarithm as such:
x log 3 = log 4
Divide both sides by log 3, and get log 4/log 3. Too bad the answer is 1 for this instance. (the other being 0, which at least works out) Can someone tell me where I have gone wrong? I have burnt 2 hours of my night trying to figure this out, and it is driving me absolutely crazy...
Thank you.
A somewhat simpler thing to do would be to multiply both sides by the denominator 2*3^x+ 3 to get 3^(2x)- 6(3^x)= -2(3^x)- 3 which we can then
write as (3^x)^2- 4(3^x)+ 3= 0, a quadratic equation in 3^x. Letting y= 3^x, we have y^2- 4y+ 3= (y- 3)(y- 1)= 0 so that y= 3^x= 3 and y= 3^x= 1, not "4 and 1".
I then add them together, and get:
(3^2x - 4 * 3^x + 3) / (2 * 3^x + 3) = 0
From here, I know that I can safely forget about the denominator, as it is only the numerator that is going to give me 0, so now I am at:
(3^2x - 4 * 3^x + 3) = 0
I put that through the quadratic formula, and find that 3^x is equal to 4, or 0. So I change this to a logarithm as such:
x log 3 = log 4
Divide both sides by log 3, and get log 4/log 3. Too bad the answer is 1 for this instance. (the other being 0, which at least works out) Can someone tell me where I have gone wrong? I have burnt 2 hours of my night trying to figure this out, and it is driving me absolutely crazy...
Thank you.
Oh my god... I see where I went wrong. I was seriously confused when I looked at what you guys posted, and everyone had the same things as me all the way until the quadratic equation. I went back, and looked at my papers, and see my problem. In the squareroot, I had 16-12, which is correct, but for some reason I didnt root it, so I had (4+-4)/2. I seriously dont see how I overlooked this so many times last night... I was so focused on it being a problem somewhere else, that I didnt look hard enough at the quadratic section I guess...
I seriously hate small errors that cost so much time! But thank you all. I had a bad nights sleep over the problem, lol.