Question

How do you solve `ln(x) + ln(x-2) = ln(x+4)`?

Answer 1

Use properties of logs to derive a quadratic equation in `x`, one of whose roots is a valid solution to the original equation.

`x = 4`

Explanation:

`ln(x+4) = ln(x) + ln(x-2) = ln(x(x-2))`

So

`x+4 = x(x-2) = x^2 - 2x`

Subtract `x+4` from both ends to get:

`0 = x^2-3x-4 = (x-4)(x+1)`

So `x = -1` or `x = 4`

Discard `x = -1` since both `ln(-1)` and `ln(-1-2)` are not defined.

So the only solution is `x = 4`