Solve for $x$ without using a calculating utility. Leave your answer in radical form, and write all fractions in lowest terms.
$\begin{matrix} \ln (\frac{7}{x}) + \ln (8 x^{3}) = \ln 5 \\ x = ◻ \end{matrix}$

Answer

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Answer

Final Answer: $x = \frac{\sqrt{70}}{28}$

Steps

Step 1 ：Rewrite the equation using the properties of logarithms: $\ln (\frac{7}{x} * 8 x^{3}) = \ln 5$

Step 2 ：Simplify the equation by multiplying the numbers inside the logarithm: $\ln (56 x^{2}) = \ln 5$

Step 3 ：Use the property that if $\ln a = \ln b$ , then $a = b$ to solve for x: $56 x^{2} = 5$

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: $x = \frac{\sqrt{70}}{28}$