Question

Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if

possible.

ⓐ ${\log }_2\frac{5}{4}$

ⓑ$\log \frac{10}{y}$

Short answer

ⓐ logarithm expression${\log }_2\frac{5}{4}$ as a difference of logarithms is${\log }_2\frac{5}{4}={\log }_{2}5-2$

ⓑ logarithm expression$\log \frac{10}{y}$ as a difference of logarithms is$\log \frac{10}{y}=1-\log y$

Step-by-step solution

Step 1. given data 

Given logarithm expressions are

ⓐ${\log }_2\frac{5}{4}$

ⓑ$\log \frac{10}{y}$

Step 2. part (a)  

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

so

$$\begin{gathered} {\log }_2\frac{5}{4}={\log }_{2}5-{\log }_{2}4 \\ {\log }_2\frac{5}{4}={\log }_{2}5-{\log }_2{2}^2 \\ {\log }_2\frac{5}{4}={\log }_{2}5-2 \end{gathered}$$

Step 3. part (b)  

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

so

$$\begin{gathered} \log \frac{10}{y}=\log 10-\log y \\ \log \frac{10}{y}=1-\log y \end{gathered}$$