4. Given $f(x)=a \sin x+b \sqrt[3]{x}+c \ln (x$ $\left.+\sqrt{x^{2}+1}\right)+4(a, b, c$ are real numbers $)$, and $f\left(\lg \log _{3} 10\right)=5$. Then the value of $f(\lg \lg 3)$ is ( ).
(A) -5
(B) -3
(C) 3
(D) varies with $a, b, c$
4. (C).

Since the function $y=\ln \left(x+\sqrt{x^{2}+1}\right)$ is an odd function, $g(x)=a \sin x+b \sqrt[3]{x}+c \ln \left(x+\sqrt{x^{2}+1}\right)$ is also an odd function, which means $f(x)-4$ is an odd function.
$$
\begin{aligned}
\text { Also, } 5 & =f(\lg \lg 10) \\
& =f\left(\lg \lg ^{-1} 3\right)=f(-\lg \lg 3),
\end{aligned}
$$

then $f(\lg \lg 3)-4=-(f(-\lg \lg 3)-4)=-1$.
Thus, $f(\lg \lg 3)=3$.