Solving modulo equations with one variable Given the following equation:
$$10 = 4^x \pmod {18}$$
How can one know what are the correct values for $x$ ?

First, $x>0$, because $x=0$ is not a solution, and $4$ has no inverse mod $18$.
Also, $2$ is a primitive root mod $9$, because by Euler's Theorem (since $\phi(9)=6$ and $\gcd(2,9)=1$) we get $2^6\equiv 1\pmod{9}$, also $2^{6/2}\equiv 8\not\equiv 1\pmod{9}$ and $2^{6/3}\equiv 4\not\equiv 1\pmod{9}$.
$$4^x\equiv 10\pmod{18}\stackrel{:2}\iff 2^{2x-1}\equiv 5\equiv 2^5\pmod{9}$$
$$\iff 2x-1\equiv 5\pmod{6}\iff 2x\equiv 6\pmod{6}$$
$$\stackrel{:2}\iff x\equiv 3\equiv 0\pmod{3}\iff x=3k$$
for some $k\in\mathbb Z^+$. This is your solution.
