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

which means
$$
f(\lg \lg 3)-4=-(f(-\lg \lg 3)-4)=-1 .
$$

Thus, $f(\lg \lg 3)=3$.