### Instruction: Find the domain of the function $a(x) = \sqrt{3^x - 7^x}.$.### Response: To find the domain of a function, I need to determine the set of values of $x$ for which the function is defined and real.
The function involves a square root, so I need to make sure that the expression inside the square root is never negative, since that would lead to imaginary numbers.
So I want to solve the inequality $3^x - 7^x \geq 0$, which means that $3^x \geq 7^x$.
To simplify this inequality, I can take the logarithm of both sides with any positive base, say base 10.
Then I get $\log_{10} 3^x \geq \log_{10} 7^x$.
Using the power rule of logarithms, I can bring the exponents down as coefficients.
Then I get $x \log_{10} 3 \geq x \log_{10} 7$.
To isolate $x$, I can subtract $x \log_{10} 3$ from both sides.
Then I get $0 \geq x (\log_{10} 7 - \log_{10} 3)$.
I can simplify the expression in parentheses using the quotient rule of logarithms.
Then I get $0 \geq x \log_{10} (7/3)$.
Since $7/3$ is greater than 1, the logarithm of $7/3$ is positive.
So I can divide both sides by $\log_{10} (7/3)$ without changing the direction of the inequality.
Then I get $0 \geq x$.
This means that the domain of the function is all the values of $x$ that are less than or equal to 0.