First, we can use Product Rule, \(\log_a(x \cdot y) = \log_a x + \log_a y\):

Next, we can use Power Rule, \(\log_a(x^n) = n \log_a x\):

Therefore, we can determine that \( \log_a (x^3y)\) expanded is \(\boldsymbol{3\log_a (x) - \log_a (y)}\).

First, we can use Product Rule, \(\log_a(x \cdot y) = \log_a x + \log_a y\):

Next, we can use Power Rule, \(\log_a(x^n) = n \log_a x\):

Therefore, we can determine that \(\log_b (x^2y^3)\) expanded is \(\boldsymbol{2\log_b (x) - 3\log_b (y)}\).

First, we can use the Quotient Rule, \(\log_a \left(\cfrac{x}{y}\right) = \log_a x - \log_a y \):

Next, we can use the Zero Rule, \(\log_a 1 = 0\):

Then, we can use Product Rule, \(\log_a(x \cdot y) = \log_a x + \log_a y\):

Therefore, we can determine that \(\log_c \left(\cfrac{1}{x^4 y^3} \right)\) expanded is \(\boldsymbol{-4\log_c (x) - 3\log_c (y)}\).

First, we can use the Quotient Rule, \(\log_a\left(\cfrac{x}{y}\right) = \log_a x - \log_a y \):

Next, we can use the Power Rule, \(\log_a(x^n) = n \log_a x\):

Therefore, we can determine that \(\log_d \left(\cfrac{x^5}{y^{1/3}} \right)\) expanded is \(\boldsymbol{5\log_d (x) - \cfrac{1}{3}\log_d (y)}\).

First, we can split \(y^6\) using the Power of a Product, \(a^{mn} = (a^m)^n\):

Next, we can use Power Rule, \(\log_a(x^n) = n \log_a x\), to the logarithms:

Then, we can simplify the expression by multiplying the coefficients:

After, we can use Quotient Rule, \( \log_a(\cfrac{x}{y}) = \log_a x - \log_a y\) to combine the logarithms:

Therefore, we can determine that \( \cfrac{1}{3}\log_e (x^3) - \cfrac{1}{2}\log_e (y^6) \) condensed is \(\boldsymbol{\log_e\left(\cfrac{x}{y^3}\right)}\).

First, we can split \((x^3 y^2)\) and \((x^4 y^3)\) using the Power of a Product, \(a^{mn} = (a^m)^n\):

Next, we can use Power Rule, \(\log_a(x^n) = n \log_a x\):

Then, we can simplify the expression by multiplying the coefficients:

After, we can use the Quotient Rule, \( \log_a(\cfrac{x}{y}) = \log_a x - \log_a y\) to combine the logarithms:

Finally, we can simplify the expression inside the logarithm by cancelling powers with the same base:

Therefore, we can determine that \( \cfrac{2}{3}\log_f (x^3 y^2) - \cfrac{1}{2}\log_f (x^4 y^3) \) condensed is \(\boldsymbol{\log_f(y^{-1/6})}\).

First, we can use the Power Rule, \(n \log_a x = \log_a(x^n)\), on the logarithms:

Next, we can combine the logarithms using Product Rule, \(\log_a(x \cdot y) = \log_a x + \log_a y\):

Therefore, we can determine that \(3\log_g (x) + \cfrac{1}{2}\log_g (y) \) condensed is \(\boldsymbol{\log_g(x^3y^{1/2})}\).

First, we can use the Power Rule, \(\log_a(x^n) = n \log_a x\), on the logarithms:

First, we can use the Power Rule, \(n \log_a x = \log_a(x^n)\), on the logarithms:

Next, we can combine the logarithms using Quotient Rule, \(\log_a\left(\cfrac{x}{y}\right) = \log_a x - \log_a y\):

Therefore, we can determine that \(3\log_h (x) - \cfrac{2}{3}\log_h (y^3)\) condensed is \(\boldsymbol{\log_h\left(\cfrac{x^3}{y^2}\right)}\).

First, we can apply the exponent to the term inside the logarithm:

Next, we can identify the power the exponential must be raised to to obtain \(729\) with a base of \(3\):

Then, we can use the Inverse Property of Logarithm since the base of the exponential is the same as the base of the logarithm:

Therefore, we can evaluate \(\log_3 (27^2)\) as equal to \(\boldsymbol{6}\).

First, we can identify the power to which the exponential must be raised to obtain \(32\) with a base of \(3\):

Next, we can use the Inverse Property of Logarithm since the base of the exponential is the same as the base of the logarithm:

Therefore, we can evaluate \(\log_2 (32)\) as equal to \(\boldsymbol{5}\).