Example 11 Solve for x
$$9 x \equiv 13(\bmod 43)$$
Solving, we look up the table at the end of this chapter and find that a primitive root of 43 can be
$$g \equiv 3(\bmod 43)$$

From the table, we also find
$$\text { ind }_{y} 9=2, \text { ind }{ }_{y} 13=32,$$

Thus, the solution is
$$\operatorname{ind}_{g} x \equiv \operatorname{ind}_{9} 13-\operatorname{ind}_{9} 9=30(\bmod \varphi(43))$$

Therefore, we have
$$\operatorname{ind}_{g} x=30,$$

Looking up the table again, we get
$$x \equiv 11(\bmod 43)$$