Given $f(x)=a \sin x-b \sqrt[3]{x}+c \ln \left(x+\sqrt{x^{2}+1}\right)+4\left(a, b, c\right.$ 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) depends on $a, b, c$
4. (C).

Since the function $y=\ln (x+\sqrt{x+1})$ is an odd function.
Therefore, $g(x)=a \sin x+b^{3} \sqrt{x}+c \ln \left(x+\sqrt{x^{2}+1}\right)$ is an odd function, which means $f(x)-4$ is an odd function.
Also, $5=f\left(\lg _{3} 10\right)=f\left(\lg ^{-1} 3\right)=f(-\lg \lg 3)$, so $f(\lg \lg 3)-4=-(f(-\lg \lg 3)-4)$ $=$ -1.
Thus, $f(\lg \lg 3)=3$.