Thread: A problem on a logarithmic curves, their tangent and their perpendicular line.

Consider the curve y=2logx, where log is the natural logarithm. Let 'l' be the tangent to that curve which passes through the origin, let 'P' be the point of contact of 'l' and that curve, and let 'm' be the straight line perpendicular to the tangent 'l' at 'P'. We are to find the equations of the straight line 'l' and 'm' and the area S of the region bounded by the curve y=2logx, the straight line 'm' and the x-axis.

Let 't' be the x-coordinate of the point P. Then 't' satisfies log t = _____? Hence the equation of 'l' is
y = (____)x/e


My question is I have no idea what the relation of the log of a x-coordinate of a point on a curve is? There may be follow up questions as I try to complete the problem but for now this is where i'm stuck at. Please help! Thanks!

Originally Posted by gundanium

Consider the curve y=2logx, where log is the natural logarithm. Let 'l' be the tangent to that curve which passes through the origin, let 'P' be the point of contact of 'l' and that curve, and let 'm' be the straight line perpendicular to the tangent 'l' at 'P'. We are to find the equations of the straight line 'l' and 'm' and the area S of the region bounded by the curve y=2logx, the straight line 'm' and the x-axis.

Let 't' be the x-coordinate of the point P. Then 't' satisfies log t = _____? Hence the equation of 'l' is
y = (____)x/e


My question is I have no idea what the relation of the log of a x-coordinate of a point on a curve is? There may be follow up questions as I try to complete the problem but for now this is where i'm stuck at. Please help! Thanks!

It should be obvious that the gradient of the line passing through O and (t, 2 ln(t)) is 2ln(t)/t. On the other hand, the gradient at (t, 2 ln(t)) is 2/t using calculus (which is why your question was moved from the PRE-calculus subforum to here, by the way). Now draw the obvious conclusion and solve for t.