## Logarithmic spiral

1. The problem statement, all variables and given/known data

a logarithmic spiral is given. The center lies on the x-axis. after a turn of 180 degrees counterclockwise I am 86.23m away from the starting point, after 360 degrees I'm 75.41m away from the start (radius, not along the spiral). Where am I after I walked exactly 3km along the spiral trajectory?

2. Relevant equations

equation: r=ae^b*theta
starting point corresponds to theta=0

3. The attempt at a solution

Um.. that's my problem. I never really learnt how to solve logarithmic calculations. Btw, this is not a homework, but part of a puzzle. I got the numbers from previous steps and found the equation online.

I understand that r=distance from origin, theta = angle with x-axis, a and b some constants. Right. But I don't see any way of working with this this equation or any of the expressions connected to it I could need a little push in the right direction.

Thanks a lot,
Martine
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Recognitions:
Hint2: $dl = \left( \sqrt{r^2 + \left( dr/d\theta \right)^2} \right) d\theta$
 Recognitions: Gold Member Science Advisor Staff Emeritus You say "180 degrees" and "360 degrees" but radian measure should be use here. when $\theta= \pi$, r= 86.23 so $86.23= ae^{b\pi}$. When $\theta= 2\pi$, r= 75.42 so $75.41= ae^{2b\pi}$. As uart says, dividing one equation by the other will cancel the 'a's leaving $$\frac{75.41}{86.23}= \frac{e^{2b\pi}}{e^{b\pi}}= e^{b\pi}$$ that is easy to solve for b. Then use that value of b with either of the first equations to find a. For the final problem you will need to integrate the "differential of arc length" that uart gave from 0 to $\theta$ and set it equal to 3000 to solve for $\theta$.