Question

How do you solve `log(2+x)-log(x-5)=log 2`?

Answer 1

`x= 12`

Explanation:

Re-write as single logarithmic expression

Note: `log(a) - log(b) = log(a/b)`

`log(2+x) - log(x-5) = log2`

`log((2+x)/(x-5))= log 2`

`10^log((2+x)/(x-5)) = 10^(log2)`

`(2+x)/(x-5) = 2`

`(2+x)/(x-5)*color(red)((x-5)) = 2*color(red)((x-5))`

`(2+x)/cancel(x-5)*cancel((x-5)) = 2(x-5)`

`2+x " " "= 2x- 10`
`+10 - x = -x +10`

===============
`color(red)(12 " " " = x)`

Check :
`log(12+2) - log(12-5) = log 2` ?
`log(14) - log(7)`
`log(14/7)`
`log 2 = log 2`

Yes, answer is `x= 12`