Find the exact length of the interval [a,b] for ln^2(x)-2ln(x)-4=0.?

Question

Find the exact length of the interval [a,b] for ln^2(x)-2ln(x)-4=0.

I need to know what to do PLEASE!!

Answer 1

ln(x)^2 - 2ln(x) - 4 = 0

same as saying

x^2 - 2x - 4 = 0

x = (-b ± sqrt(b^2 - 4ac))/(2a)

x = (-(-2) ± sqrt((-2)^2 - 4(1)(-4)))/(2(1))
x = (2 ± sqrt(4 + 16))/2
x = (2 ± sqrt(20))/2
x = (2 ± sqrt(4 * 5))/2
x = (2 ± 2sqrt(5))/2
x = 1 ± sqrt(5)

since you can't use 1 - sqrt(5), because you can't have a ln(-x)

x = 1 + sqrt(5)

now this becomes

ln(x) = 1 + sqrt(5)
inverse ln both sides

x = e^(1 + sqrt(5))

Answer 2

If this means the same as (ln(x))^2 - 2 ln(x) - 4 = 0, then it's just a quadratic equation which you can solve using the formula to get
ln(x) = 1 + or - sqrt(5).

But what is this interval [a,b]? Is it the interval in which
(ln(x))^2 - 2 ln(x) - 4 < 0 ?

In that case its exact length would be the difference between those two roots, which is 2 sqrt(5)

Answer 3

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Answer 4

just do an integration fool

Answer 5

I don't understand your question. Can you restate?

Answer 6

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