Determine a value $t_0$ such that $|u(x,t)| = |-4e^{-2t/5}\cos2x\;| < 0.0001$

Question

My problem is the following:

Determine a value $t_0$ such that $|u(x,t)| = |-4e^{-2t/5}\cos2x\;| < 0.0001,$

for $t > t_0$, with $0<x<\pi$. How to approach this problem?

According to my book the answer should be:

$$t_0 = 10\log10+\frac{5}{2}\log4\approx26.5$$

from reference:

Answer 1

Hint

The maximum vale of a cosine is $1$ as mentioned by Lutz. So, $$ |-4e^{-2t/5}\cos2x\;|< |-4e^{-2t/5}|$$ and you want this to be smaller than $0.0001$. Since the exponential term is always positive, this then reduces to $$ 4e^{-2t/5}< 10^{-4}$$. Take the logarithms of both sides and ...

Do not forget that changing sign change the sense of the inequality.

I am sure that you can take from here.

Answer 2

You are supposed to find such a t, for which, whatever x is, the said inequality holds.
Let's for a while substitute $p=4e^{\frac{-2t}{5}}$
First of all you should rewrite the equation as
$p|\cos 2x| < 0.0001$
(since $p\ge 0$)

Now let's take such $x_m$, such that $|\cos 2x_m|$ is the maximum. That would be $|\cos 2x_m|=1$
Now you have $p < 0.0001$ which can be easily solved using logarithms.

For any other x, $|\cos 2x| \le |\cos 2x_m|$ and thus $p|\cos 2x| \le p|\cos 2x_m| < 0.0001$