Question

How do you solve the logarithmic equation `ln(x) + ln(x - 1) = ln2`?

Answer 1

Derive a quadratic equation, one of whose roots is a solution of the original equation, namely `x=2`.

Explanation:

Note first that we want `x > 0` and `x - 1 > 0` in order that the natural logarithms are well defined.

If that is so, then we can say:

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

Then since `ln` is a one-one function from `(0, oo) -> RR` we require that:

`x(x-1) = 2`

Rearrange this to get:

`x^2-x-2 = 0`

which has roots `-1` and `2`

The value `x=-1` is not an acceptable solution of the original equation, since we want `x > 0`. So the only acceptable solution is `x=2`, which does work:

`ln(2) + ln(2 - 1) = ln(2) + ln(1) = ln(2) + 0 = ln(2)`