Question

In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
${\log }_3\left(\sqrt{4}{x}^7{y}^8\right)$.

Short answer

The solution is${\log }_3\left(2\right)+7{\log }_3\left(x\right)+8{\log }_3\left(y\right)$.

Step-by-step solution

Step 1. Given and explanation.

We have ${\log }_3\left(\sqrt{4}{x}^7{y}^8\right)$.

The product property states that if there is an expression as ${\log }_a(m·n)$, it is equal to ${\log }_{a}m+{\log }_{a}n$.

Applying this, we get the solution.

Step 2. Finding the solution.

We have ${\log }_3\left(\sqrt{4}{x}^7{y}^8\right)$.

Applying the product property, we get
$$\begin{gathered} ={\log }_3\left(\sqrt{4}{x}^7{y}^8\right) \\ ={\log }_3\left(\sqrt{4}\right)+{\log }_3{\left(x\right)}^7+{\log }_3{\left(y\right)}^8 \\ ={\log }_3\left(2\right)+7{\log }_3\left(x\right)+8{\log }_3\left(y\right) \end{gathered}$$
Thus we get the value.