Question

Use the Properties of Logarithms to condense the logarithm ${\log }_{2}5+{\log }_{2}x-{\log }_{2}y$. Simplify, if possible.

Short answer

The condensed form of the logarithm is${\log }_{2}5+{\log }_{2}x-{\log }_{2}y={\log }_2\frac{5x}{y}$

Step-by-step solution

Step 1. given data   

Given logarithm expressions is${\log }_{2}5+{\log }_{2}x-{\log }_{2}y$

Step 2. Apply product Property of the logarithm 

Use product Properties of Logarithm ${\log }_{a}m+{\log }_{a}n={\log }_a\left(m·n\right)$

role="math" localid="1644961031838" ${\log }_{2}5+{\log }_{2}x-{\log }_{2}y={\log }_{2}5x-{\log }_{2}y$

Step 3. apply Quotient Property of logarithm

Use Quotient Property of the logarithm${\log }_a\left(m\right)-{\log }_a\left(n\right)={\log }_a\left(\frac{m}{n}\right)$

$$\begin{gathered} {\log }_{2}5+{\log }_{2}x-{\log }_{2}y={\log }_{2}5x-{\log }_{2}y \\ {\log }_{2}5+{\log }_{2}x-{\log }_{2}y={\log }_2\frac{5x}{y} \end{gathered}$$