The argument of a logarithm function must always be positive, so we have the restrictions x > 0 2x - 1 > 0, equivalent to x > 1/2 x - 2 > 0, equivalent to x > 2. This is the most restrictive condition, so it governs. Taking the anti-log of both sides of the equation, we have x = (2x-1)/(x-2) x(x-2) = 2x-1 (multiply both sides by x-2) x^2 - 2x = 2x - 1 (use the distributive property to eliminate parentheses) x^2 - 4x + 1 = 0 (subtract 2x-1 from both sides to put the equation into standard form) x = (-(-4) ±√((-4)^2 - 4(1)(1)))/(2(1)) (apply the quadratic formula) x = (4 ±√(16-4))/2 (simplify) x = (4 ±2√3)/2 (simplify) x = 2±√3 (simplify) Since we know that x must be greater than 2, only the positive sign will give a suitable solution.

##### x = 2+√3 ≈ 3.732

__Check__Ln[3.732] = Ln[2*3.732-1] - Ln[3.732 - 2] 1.317 = Ln[7.464 - 1] - Ln[1.732] 1.317 = Ln[6.464] - .549 1.317 = 1.866 - .549 (yes)