Solve the logarithmic equation lnx - ln(x+1) = 2?

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For the beginning, we'll use the quotient property of the logarithms:

lnx - ln(x+1) = ln [x/(x+1)]

Now, we'll have to use the one to one property, that means that:

ln [x/(x+1)] = 2 lne if and only if [x/(x+1)] = e^2

After cross multiplying, we'll get:

x = x*e^2 + e^2

We'll move the terms which are containing the unknown, to the left side:

x - x*e^2 = e^2

After factorizing, we'll get:

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

x = e^2/(1-e^2)

But 1-e^2<0, so x = e^2/(1-e^2)<0, which is impossible because x has to be positive!

So, the equation has no solutions.

ln x -ln (x+1) = 2

since ln x-ln y = ln x/y

then ln(x/(x+1)= 2

==> e^2 = x/(x+1)

==> (x+1)e^2 = X

==> xe^2 + e^2 = x

==> xe^2 -x= -e^2

==> x(e^2-1) = - e^2

==> x = -e^2/(e^2 -1)

To solve: lnx-ln(x+2) = 2.

Solution:

LHS = ln {x/(x+2)} = 2. Taking anti logarithms on both sides,

x/(x+2) = e^2. Or

x = e^2(x+2) . Or

x-xe^2 = 2e^2. Or

x(1-e^2) = 2e^2. Or

x = 2e^2/(1-e^2)

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