# 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!

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

• Nice encouragement and assistance! +1 Nov 18, 2013 at 14:33
• Thanks Amy. Have a wonderful day ahead.:-) Nov 18, 2013 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?

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

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

$$2 = 999x$$

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