4. The equation $3^{x}+5^{x}+7^{x}=11^{x}$ has $(\quad)$ distinct real roots.
(A) 0
(B) 1
(C) 2
(D) 3
4. B.

Let $f(x)=\left(\frac{3}{11}\right)^{x}+\left(\frac{5}{11}\right)^{x}+\left(\frac{7}{11}\right)^{x}-1$.
Then $3^{x}+5^{x}+7^{x}=11^{x} \Leftrightarrow f(x)=0$.
Notice that, $f(0)=2>0, f(2)=-\frac{38}{121}<0$.
Therefore, $f(x)=0$ has a real root in the interval $(0,2)$.

Since $f(x)$ is strictly monotonically decreasing on $(-\infty,+\infty)$, the equation $f(x)=0$ has exactly one real root.