Solve for x the following equation logx (x+1)=2; x is the base of logarithm.

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We'll impose the constraints oif existence of logarithm:

x belongs to (0;+inf.)-{1}

Now, we'll solve the equation taking antilogarithm:

x + 1 = x^2

We'll use the symmetrical property and we'll shift all terms to one side:

x^2 - x - 1 = 0

We'll apply quadratic formula:

x1 = [1 + sqrt(1 + 4)]/2

x1 = (1+sqrt5)/2

x2 = (1-sqrt5)/2

**Since the second value of x does not belong to the interval of admissible solutions, therefore we'll keep only a single value of x, namely x = (1+sqrt5)/2.**

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