Question: Find the value of the function $y = f(x)$ for the given value of $x$. Write "undefined" if the value of $x$ is not in the domain of the function.
$f(x) = 7\log_x^3$, $x = 343$
1. **State the problem:** We need to find the value of the function $$f(x) = 7 \log_x 3$$ when $$x = 343$$.
2. **Recall the logarithm definition:** $$\log_a b$$ means the exponent to which the base $$a$$ must be raised to get $$b$$.
3. **Check the domain:** The base $$x$$ of a logarithm must be positive and not equal to 1. Here, $$x = 343$$ which is positive and not 1, so the function is defined.
4. **Rewrite the logarithm:** We want to find $$\log_{343} 3$$.
5. **Express 343 as a power:** $$343 = 7^3$$.
6. **Use change of base formula:**
$$\log_{343} 3 = \frac{\log_7 3}{\log_7 343}$$
7. **Calculate denominator:**
$$\log_7 343 = \log_7 7^3 = 3$$
8. **Calculate numerator:** $$\log_7 3$$ remains as is since 3 is not a power of 7.
9. **Substitute back:**
$$\log_{343} 3 = \frac{\log_7 3}{3}$$
10. **Calculate function value:**
$$f(343) = 7 \times \log_{343} 3 = 7 \times \frac{\log_7 3}{3} = \frac{7}{3} \log_7 3$$
**Final answer:** $$f(343) = \frac{7}{3} \log_7 3$$