Question

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

$4{\log }_{2}x+6{\log }_{2}y$

Short answer

On simplifying the expression we get:${\log }_2{x}^4{y}^6$

Step-by-step solution

Step 1. Given information

The given expression is $4{\log }_{2}x+6{\log }_{2}y$

We have to condense the expression using logarithm properties.

Step 2. Identify the logarithm property

Use the power property of logarithm:

${\log }_a{\left(M\right)}^p=p{\log }_a\left(M\right)$

Step 3. Use the power property of logarithm

$$\begin{gathered} 4{\log }_{2}x+6{\log }_{2}y \\ ={\log }_2{x}^4+{\log }_2{y}^6 \end{gathered}$$

Step 4. Use the product property of logarithm

Use product property of logarithm which states:

${\log }_{a}M+{\log }_{a}N={\log }_a\left(M·N\right)$

Step 5. Apply the product property of logarithm

$$\begin{gathered} {\log }_2{x}^4+{\log }_2{y}^6 \\ ={\log }_2{x}^4{y}^6 \end{gathered}$$