Solving Logarithmic Spirals for Length from Origin to Point X

[eXmag]

Homework Statement

This is a logarithmic spiral. How can I find the length of the spiral (curve) from the origin to point X on the curve? The distance between the origin and the point is given (we can call that the radius) and the distance between the origin and point A is given which this is the same between points A and B and C. Point X is on the spiral with an indicated arrow pointing towards it. The location, therefore, the angle between the origin and this point is unknown. However the distance is known. How can I find the total length of the spiral from the origin to point X?

Mod note: The text above is from the oversized image originally posted. A cropped version of the image is below.[/color]

Homework Equations

1. $r=ae^{b\theta}$
2. $L=r\theta$

The Attempt at a Solution

Do I need to solve for theta then use L=r(theta) to find the total length?
I'm having troubles finding the terms for the first equation.

 

[mfb]

Do I need to solve for theta then use L=r(theta) to find the total length?

That is a good idea.

Im having troubles finding the terms for the first equation.

You know that A and B satisfy the equation for the spiral. This allows to determine a and b.

 

[eXmag]

Are you referring to the points A and B? So my values at point A and B correspond to the a and b terms in the equation?

 

[mfb]

Are you referring to the points A and B?

Sure

So my values at point A and B correspond to the a and b terms in the equation?

No. The names are a bit misleading in that respect.