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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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  • $\begingroup$ Nice encouragement and assistance! +1 $\endgroup$ – amWhy Nov 18 '13 at 14:33
  • $\begingroup$ Thanks Amy. Have a wonderful day ahead.:-) $\endgroup$ – mrs Nov 18 '13 at 14:36
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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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$\dfrac{x+2}{x} = 1000$

Dividing both the terms in numerator by x

$\dfrac{x}{x} + \dfrac{2}{x} = 1000$

$1 + \dfrac 2x = 1000$

Subtracting 1 from both side

$1 - 1 + \dfrac 2x = 1000 -1$

$\dfrac 2x = 999$

Multiplying both side by x

$\dfrac 2x \cdot x= 999x$

x cancel out leaving just 2 on left hand side and 99x on right hand side

$2 = 999x$

Therefore the final answer is

$$x = \dfrac{2}{999}$$

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  • 1
    $\begingroup$ 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 $\endgroup$ – Amzoti Nov 20 '14 at 18:17

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