Use the properties of logarithms to expand the following expression. logsqrt ((yz^5)/(x^3)) Each logarithm should involve only one variable and should not have any radicals or exponents. You may assume that all variables are positive. logsqrt ((yz^5)/(x^3))=

To expand the given expression using the properties of logarithms, we can follow these steps:<br /><br />1. Rewrite the expression inside the logarithm as a product of variables and exponents.<br />2. Apply the logarithm properties to separate the terms.<br /><br />Given expression: $log\sqrt {\frac {yz^{5}}{x^{3}}}$<br /><br />Step 1: Rewrite the expression inside the logarithm as a product of variables and exponents.<br />$\sqrt {\frac {yz^{5}}{x^{3}}} = \frac {y^{1/2}z^{5/2}}{x^{3/2}}$<br /><br />Step 2: Apply the logarithm properties to separate the terms.<br />$log\sqrt {\frac {yz^{5}}{x^{3}}} = log\left(\frac {y^{1/2}z^{5/2}}{x^{3/2}}\right)$<br />$= log(y^{1/2}) + log(z^{5/2}) - log(x^{3/2})$<br />$= \frac{1}{2}log(y) + \frac{5}{2}log(z) - \frac{3}{2}log(x)$<br /><br />Therefore, the expanded expression is:<br />$log\sqrt {\frac {yz^{5}}{x^{3}}} = \frac{1}{2}log(y) + \frac{5}{2}log(z) - \frac{3}{2}log(x)$