# Solving $\log(x+2) - \log(x) = 3$

I have work through the whole problem, but I cannot get passed the last step.

The original equation was: $\log(x+2) - \log(x) = 3$

I worked it out to this: $\frac{x+2}{x} = 1000$.

I know the answer is $\frac{2}{999}$ but I don't know how to get there. It's probably really simple, but I am just drawing a blank! Any help would be just great, thanks!

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You are almost there!. While: $$\frac{x+2}{x}=1000\to x+2=1000x\to 1000x-x=2\to 999x=2\to x=\frac{2}{999}$$

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Nice encouragement and assistance! +1 –  amWhy Nov 18 '13 at 14:33
Thanks Amy. Have a wonderful day ahead.:-) –  Babak S. Nov 18 '13 at 14:36

Multiply both sides by $x$ to find

$$x + 2 = 1000x$$

Subtract $x$ from each side to get

$$2 = 1000x - x$$

Can you take it from there?

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(x+2)/x = 1000
Dividing both the terms in numerator by x

--> (x/x) + (2/x) = 1000 --> 1 + (2/x) = 1000 Subtracting 1 from both side --> 1 -1 + (2/x) = 1000 -1 --> 2/x = 999 Multiplying both side by x --> (2/x)*x = 999 *x x cancel out leaving just 2 on left hand side and we get 99x on right hand side --> 2 = 99x Therefore the final answer is --> x = 2/999

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Welcome to MSE! It really helps readability to use MathJax (please see FAQ). I am also not sure how this differs from the accepted answer. Regards –  Amzoti Nov 20 '14 at 18:17