Question

determine the equation of the tangent to the curve defined by y=ln x -1 that is parallel to the straight line with the equation 3x-6y-1=0

Answer 1

First of all you need to know to know the slope of the line you're required to be parallel to. By solving your second equation for you get a slope m.
Knowing that the first order derivative of a function in a point represent the slope of the tangent to the line in the point, you calculate the derivative of your y=ln x -1 and set it = m. By solving y'=m you get the x of the point your tangent passes by. It's y is taken by replacing the found x in the first equation. Then you know the slope and one point, which are enough to determine your straight line.