3. First prove the inequality $\left(1+b_{1} x\right)\left(1+b_{2} x\right) \cdots\left(1+b_{n-1} x\right)\left(F+b_{n} x\right) \geqslant$ $\left[1+\sqrt[n]{b_{1} b_{2} \cdots b_{n}^{=} x}\right]^{\underline{n}}$.

Consider $f(x)=\lg \left(1+10^{x}\right)$, by Jensen's inequality $\frac{f\left(x_{1}\right)+f\left(x_{2}\right)+\cdots+f\left(x_{m}\right)}{m} \geqslant$
$$f\left[\frac{x_{1}+x_{2}+\cdots+x_{m}}{m}\right]$$

So
$$f(x) \leqslant(\sqrt[n+1]{a}+\sqrt[n+1]{b})^{n+1}$$