How to Find the Intersection of a Logarithmic Curve and a Tangent Line?

[AngleWyrm]

TL;DR Summary

Trying to find coordinates of an intersection between a tangent line and a curve

I have a formula y=log(x)/log(0.9) which has this graph:

I want to find the intersection of this curve and a tangent line illustrated in this rough approximation:

The axes have very different scales, so the line isn't actually a slope of -1, it's just looks that way.

How can I figure out:
1). the actual slope of the line
2). the coordinates of the intersection

 

[anuttarasammyak]

Why do don't you calculate dy/dx to get the slope of the tangential line ?

 

[AngleWyrm]

Yeah that works; looks like about -25/0.5 = -50 slope

 

[pbuk]

How can I figure out:
1). the actual slope of the line
2). the coordinates of the intersection

If you know one of these you can determine the other, but if you don't know either then the problem is not well defined: there are an infinite number of tangent lines with some slope that intersect the curve somewhere.

 

[Mark44]

Summary:: Trying to find coordinates of an intersection between a tangent line and a curve

I want to find the intersection of this curve and a tangent line illustrated in this rough approximation:

In the lower of your two graphs, it looks like you are concerned with the point (.2, f(.2)). Is this the point of intersection for this problem? If so, that is information you should have provided.

 

[AngleWyrm]

it looks like you are concerned with the point (.2, f(.2)). Is this the point of intersection for this problem?

No, that's an estimate of the answer to the question what are the coordinates.

 

[Mark44]

No, that's an estimate of the answer to the question what are the coordinates.

Are you given the x-coordinate at the point of intersection? This is post #7, and we're still not sure exactly what you're asking.

If not, the problem is not well defined, as @pbuk said.

If you know one of these you can determine the other, but if you don't know either then the problem is not well defined: there are an infinite number of tangent lines with some slope that intersect the curve somewhere.

 

[AngleWyrm]

Yeah that works; looks like about -25/0.5 = -50 slope

If you know one of these you can determine the other

You know one of these. Carry on.

 

[Mark44]

From post #5:

it looks like you are concerned with the point (.2, f(.2)).

Your reply:

No, that's an estimate of the answer to the question what are the coordinates.

Is x = .2 the known x-coordinate at the point of tangency? If so, the point (.2, f(.2)) is NOT an estimate.

You know one of these.

No we don't! You haven't said what it is that is known in this problem.
I'm about to give up here and lock this thread, unless you are more forthcoming with the information of this problem.

 

[anuttarasammyak]

Do you want one of these lines ?

 

[robphy]

As others have said, there are an infinite number of tangent lines.
https://www.desmos.com/calculator/ddfh2snwjy

 

[AngleWyrm]

Is x = .2 the known x-coordinate

Re-read the thread summary

As others have said, there are an infinite number of tangent lines.

Are there an infinite number of tangent lines with a slope of -50?
Or does that restrict the answer set to one unique line and set of coordinates?

 

[anuttarasammyak]

Re-read the thread summaryAre there an infinite number of tangent lines with a slope of -50?
Or does that restrict the answer set to one unique line and set of coordinates?

Say the condition is -50 slope, the solution of the equation
$$y'(x_0)=-50$$
gives unique tangential point $(x_0,y_0)=(x_0,\frac{\log x_0}{\log 0.9})$ and thus the equation of the line is
$$y-y_0=y'(x_0)(x-x_0)=-50(x-x_0)$$

 

[Mark44]

Is x = .2 the known x-coordinate at the point of tangency?

Re-read the thread summary

I have read it several times. I've also asked you several times to tell us what information is given in this problem. Since you refuse to do so, I am closing this thread.

 

[jedishrfu]

Looks like it's time to close this thread. While the problem is missing some key info or the OP is unwilling to provide it, the only possible solution is to read an approximated slope from the given graph by inspection.

Basically, to find the tangent line accurately for a known algebraic function, we need to know the function and the x point or y point of the tangent line of interest. While we have the equation for the curve, we don't know what x point or y point to use.