Question

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

Answer 1

Firs property used : Quotient Property

`log_b((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)^{(1)/(n)} = (1)/(n) log_b X`

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

Fully expanded form:

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