Question

How do you solve `log(x+1) - log(x-1)=1`?

Answer 1

Use the law

`log(a) - log(b) = log(a/b)`.

Hence

`log((x+ 1)/(x- 1)) = 1`

`(x + 1)/(x - 1) = 10`

`x +1 = 10(x - 1)`

`x + 1 = 10x - 10`

`11 = 9x`

`x = 11/9`

We now verify that it satisfies the equation. It will as long as `x > 1`, which it is. Hence, we're done.

Hopefully this helps!

Answer 2

The solution is `x=11/9`.

Explanation:

Use these `log`arithm rules to help simplify the equation:

`log(color(red)x)color(green)-log(color(blue)y)=log(color(green)(color(red)x/color(blue)y))`

`log(x)=log(y)qquadquad=>qquadquadx=y`

Now here's the actual problem:

`log(color(red)(x+1))color(green)-log(color(blue)(x-1))=1`

`log(color(green)(color(red)(x+1)/color(blue)(x-1)))=1`

`log(color(green)(color(red)(x+1)/color(blue)(x-1)))=log(10)`

`color(green)(color(red)(x+1)/color(blue)(x-1))=10`

`color(red)(x+1)=10(color(blue)(x-1))`

`color(red)(x+1)=10x-10`

`color(red)x+11=10x`

`11=9x`

`11/9=x`

`x=11/9`

That's the solution. Hope this helped!