When two lines meet!! 4

For start, let's consider graph of each

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Graph of $log_{10}\sqrt{|x|}$

Graph of $cos(sin(x))$

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So, we can see that $cos(sin(x))\in [cos(1), 1]$

So, we have to find for what values of $x$ , $log_{10}\sqrt{|x|} \in [cos(1), 1]$

$cos(1) \leq log_{10}\sqrt{|x|} \leq 1$

$10^{cos(1)} \leq \sqrt{|x|} \leq 10$

$10^{2cos(1)} \leq |x| \leq 100$

$12 \leq |x| \leq 100$

$4\pi \leq |x| \leq 31.83\pi$

As it is mentioned for $|x| \leq 31π$

$4\pi \leq |x| \leq 31\pi$

$-31\pi \leq x \leq -4\pi \text{ or } 4\pi \leq x \leq 31\pi ........... (1)$

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Now, for each $x \in [n\pi, (n+1)\pi]$ in $(1)$ , there are $\boxed{2}$ intersections

So, there are a total of $\color{#3D99F6}{\boxed{((31 - 4) × 2) × 2 = 108}}$ intersections

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Here is a graph combining both equations