Q:

Logarithmic Problem Solution: Physics or Math?

Someone have me this problem claiming that I only need to use logs to solve it, though it seems to have a physics origin:
If a coil spring is stretched 1.5 m beyond its resisting point and then released it will return to a point which is 90 of the previous distance from the resting point. How many vibrations are required before the spring moves less tan 10 cm from its resting point?
I do not have much of a physics background, but could someone explain to me how this converts into a log equation? I tried myself several avenues (mainly of the form a = blog(c)^d) but nothing seems to work

A:

Let's think about discrete time step $t$ and let $x_t$ be the position of the end of the spring on the horizontal $x$-axis after $t$ oscillation. We suppose the resting point to be at $x=0$.
At start we have $x_0 = 150$, then after one osillation, it will go on the other side of the resting point and loose a bit of strength, that is $x_1 = -0.9 \cdot 150$, then go on the other side again: $x_2 = 150 \cdot 0.9 \cdot 0.9$.
We have the recurrence relation $x_t = -0.9 x_{t-1}, \ x_0 = 150$. This can be solve easily $x_t = (-1)^t 150\cdot 0.9^t$.
Now if we only care about "being less than 10cm from the resting point", we can consider only the absolute value of $x_t$ and get rid of the alternating sign.
Hence we look for
$$
150\cdot 0.9^t \leq 10 \implies 0.9^t \leq \frac{1}{15} \implies t\log(0.9) \leq -\log(15)
$$
I let you finish. (And remember that $\log(0.9) < 0$, this is important when dealing with inequalities.)

