Pre-calculus Question: How to solve ln(x-1)+ln(x+2)=1

It seems that there are two ways to solve this question and that they don't give the same answers.
First way:

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

ln(x-1)(x+2)=1 Laws of Logarithms

(x-1)(x+2)=e

x^2+x-2-e=0

Quadratic Equation yields:

Something annoying. See below.


The second way:

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

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

ln(x-1)(x+2)-ln(e)=0 ln(e)=1

ln(x-1)(x+2)/ln(e)=0 Laws of Logarithms

ln(x-1)(x+2)=0 ln(e)=1

(x-1)(x+2)=e^0=1

x^2+x-2=1

x^2+x-3=0

Quadratic Equation yields:

[-1 +/- 13^(1/2) ]/2

So what went wrong? Is it not all right to turn 1 into whatever I would like it to be?
Thanks for your help. :)

The first method is right.

In the second method, the mistake is

ln(x-1)(x+2) - ln(e) = 0 is NOT EQUAL to ln(x-1)(x+2)/ ln(e)

The correct law of logarithm is ln a - ln b = ln (a/b) BUT NOT ln(a) / ln(b)

If you correct that mistake both methods yield same result.

In the first method, you had an error in the multiplication of the factors:
(x - 1)(x + 2) - e= x^2 -x +2x -2 = x^2 +x -2 -e (You had -1 at the end before the e)

Then, there is an error in your substitution the second time.

The second way:

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

ln[(x-1)(x+2)]-1=0 <--- notice that the ln is of the product, first of all

ln(x-1)(x+2)-ln(e)=0 ln(e)=1

ln(x-1)(x+2)/ln(e)=0 <--- this would be inside the ln when you take the next step
ln [[(x-1)(x+2)]/e] = 0

Now, you can take e of both sides since you only have an ln term on the left to get
[(x-1)(x+2)]/e = 1

Notice that you can simplify by multiplying e on both sides to get
(x-1)(x+2)=e
To solve the quadratic, subtract the e on both sides and you are back to your equations in "the first way"
x^2 +x -1 -e = 0
Yep, it is messy, sorry.

You went wrong here:

ln(x-1)(x+2)/ln(e)=0

ln(a) - ln(b) is ln(a / b), not ln(a) / ln(b)


ln((x - 1) * (x + 2)) - ln(e) =>
ln((x - 1) * (x + 2) / e)

ln((x - 1) * (x + 2) / e) = 0
(x - 1) * (x + 2) / e = e^0
(x - 1) * (x + 2) / e = 1
(x - 1) * (x + 2) = e^1
(x - 1) * (x + 2) = e
x^2 - x + 2x - 2 = e
x^2 + x - 2 - e = 0
x = (1 +/- sqrt(1 + 8 + 4e)) / (2)
x = (1 +/- sqrt(9 + 4e)) / 2

ln(x-1)(x+2)-ln(e)=0 ln(e)=1
ln(x-1)(x+2)/ln(e)=0 Laws of Logarithms

This step is wrong. You need to go o ln(x-1)(+2)/(e) not ln(e)