Example 4 Let $f(x)=\lg \frac{1+2^{x}+4^{x} a}{3}, a \in \mathbf{R}$, if when $x \in(-\infty, 1]$, $f(x)$ is meaningful, find the range of values for $a$.

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From the problem, we know that $\frac{1+2^{x}+4^{x} a}{3}>0$ always holds when $x \in(-\infty, 1]$, which means
$$
a>\frac{-1-2^{x}}{4^{x}}=-\left(\frac{1}{4}\right)^{x}-\left(\frac{1}{2}\right)^{x} .
$$

Let $\mu=-\left(\frac{1}{4}\right)^{x}-\left(\frac{1}{2}\right)^{x}, x \in(-\infty, 1]$, then $a>(\mu)_{\text {max }}$. Also, $\mu_{1}=-\left(\frac{1}{4}\right)^{x} ; \mu_{2}=$ $-\left(\frac{1}{2}\right)^{x}, x \in(-\infty, 1]$ are both increasing functions, so $\mu=-\left(\frac{1}{4}\right)^{x}-\left(\frac{1}{2}\right)^{x}, x \in(-\infty, 1]$ is an increasing function, $\mu(x)_{\max }=\mu(1)=-\frac{3}{4}$, hence $a>-\frac{3}{4}$.

Key points for solving the problem: When discussing the constant validity of functions with parameters, separating the parameters is the most commonly used method; pay attention to the use of function monotonicity when finding the maximum or minimum value.