Use properties of logarithms to expand the logarithmic expression as much as possible. Assume a and b are positive real numbers. \( \log \sqrt[4]{\frac{a}{b}} \) \( \log ^{4} \sqrt{\frac{a}{b}}=\square \) (Type an exact answer in simplified form.)

To expand the logarithmic expression \( \log \sqrt[4]{\frac{a}{b}} \), we can start by rewriting the expression in exponential form. First, we can express the fourth root as a power: \[ \log \sqrt[4]{\frac{a}{b}} = \log \left(\frac{a}{b}\right)^{1/4} \] Using the power property of logarithms, which states that \( \log(x^n) = n \log(x) \), we have: \[ \log \left(\frac{a}{b}\right)^{1/4} = \frac{1}{4} \log \left(\frac{a}{b}\right) \] Next, we can expand using the quotient property of logarithms, which is \( \log \left(\frac{x}{y}\right) = \log x - \log y \): \[ \frac{1}{4} \log \left(\frac{a}{b}\right) = \frac{1}{4} (\log a - \log b) \] Putting it all together, we arrive at the expanded form: \[ \log \sqrt[4]{\frac{a}{b}} = \frac{1}{4} \log a - \frac{1}{4} \log b \] So the final answer is: \[ \log \sqrt[4]{\frac{a}{b}} = \frac{1}{4} \log a - \frac{1}{4} \log b \]