Solve for $x$ without using a calculating utility. Leave your answer in radical form, and write all fractions in lowest terms.
\[
\begin{array}{c}
\ln \left(\frac{7}{x}\right)+\ln \left(8 x^{3}\right)=\ln 5 \\
x=\square
\end{array}
\]
Answer
Final Answer: \(x = \boxed{\frac{\sqrt{70}}{28}}\)
Step 1 ：Rewrite the equation using the properties of logarithms: \(\ln \left(\frac{7}{x} * 8x^3\right) = \ln 5\)
Step 2 ：Simplify the equation by multiplying the numbers inside the logarithm: \(\ln (56x^2) = \ln 5\)
Step 3 ：Use the property that if \(\ln a = \ln b\), then \(a = b\) to solve for x: \(56x^2 = 5\)
Step 4 ：Solve the equation for x: \(x = \pm\sqrt{\frac{5}{56}}\)
Step 5 ：Since x is in the denominator of the original equation, x cannot be negative. Therefore, the only valid solution is \(x = \sqrt{\frac{5}{56}}\)
Step 6 ：Final Answer: \(x = \boxed{\frac{\sqrt{70}}{28}}\)
