Question

How do you solve `ln x + ln(x-2) = 1`?

Answer 1

`1+sqrt(1+e)~~2.9283`

Explanation:

Use the properties of logarithms to rewrite the expression.

`ln(x(x-2))=1`

Rewrite both sides in terms of the base `e`

`e^(ln(x(x-2)))=e^1`

From the properties of logarithms this becomes

`x(x-2)=e`

Distribute `x` on left hand side

`x^2-2x=e`

Subtract `e` form both sides

`x^2-2x-e=0`

Applying the quadratic formula

`(-(-2)+-sqrt((-2)^2-4(1)(-e)))/2(1)`

`(2+-sqrt((4+4e)))/2`

`(2+-sqrt(4(1+e)))/2`

`2/2+-(2sqrt(1+e))/2`

`1+-sqrt(1+e)`

We are only interested in `1+sqrt(1+e)~~2.9283`

The other root is negative and not in the domain of the logarithmic function.