How do you solve for x: ln(x-1) + ln(x+2) = 1

Question

Here's what I did:

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

x^2 + x - 2 = e and I don't know what to do afterwords. Could you show me the steps?

Answer 1

hi,

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

Here's what I did:

x^2 + x - 2 = e completely right

Now, you need to choose one method to solve quadratic equation ( I chose quadratic formula)

x^2 + x - 2 - e = 0 a= 1 b = 1 c=-2-e Δ= b²-4ac = 1 - 4 (1)(-2-e) =9 +4e

x₁,x₂ = ( -b ±√Δ ) /2a = ( -1 ±√9+4e )/2

now your answer is x₁ = ( -1 +√9+4e )/2 AND x₂ = ( -1 -√9+4e )/2 that only x₁is acceptable.

you know when you solve equation include logarithm you need to check your answers on original equation that front of log must be positive. By doing that, x₁

make positive (x-1) ,(x+2), so only x₁ is acceptable. as a result this equation has only one answer such as

x₁ = ( -1 +√9+4e )/2

I hope it was useful.

Minoo

Answer 2

You got as far as

x² + x - 2 = e

or

x² + x - 2 - e = 0 <--- Let's concentrate on this way of saying it

which is hard to solve for x because the x appears in this equation in two places.

We can make the x appear in only one place by "completing the square".

If the left side of this equation had been x² + x + 1/4 instead of x² + x - 2 - e, we could have factored the left side into two equal factors:

x² + x + 1/4 = (x + 1/2)(x + 1/2)

So let's change the left side into x² + x + 1/4 by adding 9/4 + e to both sides of the equation x² + x - 2 - e = 0:

x² + x + 1/4 = 9/4 + e

Now factor the left side:

(x + 1/2)(x + 1/2) = 9/4 + e

Now take square roots on both sides, so that x appears in only one place:

x + 1/2 = ±√(9/4 + e) Remember to write the ± symbol.

Now subtract 1/2 from both sides to get the x all by itself:

x = ±√(9/4 + e) - 1/2 At this stage it looks as if there are two possible answers because of the ±

But remember, the original problem talks about ln(x-1), and you can't take the natural logarithm of a non-positive number, so x has to be greater than 1.

Therefore we have only one possible answer, x = √(9/4 + e) - 1/2, which is positive.