Question

Explain which properties you would use to fully expand (the problem in the screenshot)

Answer 1

Answer:

Firs property used : Quotient Property

           $log_b(\frac{X}{Y} ) = log_bX -log_bY$

   $log_b(\frac{\sqrt[3]{x} }{36y^2} ) = log_b(\sqrt[3]{x}) - log_b(36y^2)$

Second property used : Product Property

          $log_b(XY) = log_bX + log_b Y$

    $log_b(\sqrt[3]{x}) - log_b(36y^2) = (log_b(\sqrt[3]{x}) - [log_b36 + log_by^2)$

Third property used : Power Property

                 $log_b \sqrt[n]{X} = log_b(X)^{\frac{1}{n}} = \frac{1}{n} log_b X$

 $(log_b(\sqrt[3]{x}) - [log_b6^2 + log_by^2) = \frac{1}{3} log_bx - 2log_b 6 - 2log_by$

Fully expanded form:

                   $log_b(\frac{\sqrt[3]{x} }{36y^2} ) = \frac{1}{3} log_b x-2log_b6-2log_by$