Question - Complex Logarithmic Expression Simplification

\[\text{Given expression: } \frac{\log 343}{1 + \frac{1}{2}\log(49) - \frac{1}{3}\log\left(\frac{1}{125}\right)}\]

\[= \frac{\log 7^3}{1 + \frac{1}{2}\log(7^2) - \frac{1}{3}\log\left(5^{-3}\right)}\]

\[= \frac{3\log 7}{1 + \frac{1}{2}\cdot 2 \log 7 - \frac{1}{3} \cdot (-3) \log 5}\]

\[= \frac{3\log 7}{1 + \log 7 + \log 5}\]

\[= \frac{3\log 7}{\log 7 + \log 5 + \log 10}\]

\[= \frac{3\log 7}{\log(7 \cdot 5 \cdot 10)}\]

\[= \frac{3\log 7}{\log 350}\]

\[\text{Since } \log_a b = \frac{1}{\log_b a}, \text{ swap the logs:}\]

\[= \frac{3}{\log_{7} 350}\]

\[= \frac{3}{\log_{7} 7^2 + \log_{7} 5}\]

\[= \frac{3}{2 + \log_{7} 5}\]

\[\text{Without information on the value of } \log_{7} 5, \text{ this is the simplest form.}\]