Question

In the following exercises, use properties of logarithms to write each expression as a sum of logarithms, simplifying if possible

${\log }_2\sqrt[4]{\frac{5x^3}{16y^2{z}^7}}$

Short answer

The value is$\frac{1}{4}{\log }_{2}5+\frac{3}{4}{\log }_{2}x-1-\frac{1}{2}{\log }_{2}y-\frac{7}{4}{\log }_{2}z$

Step-by-step solution

Step 1. Given

An expression is given as${\log }_2\sqrt[4]{\frac{5x^3}{16y^2{z}^7}}$

Step 2. Calculation

Rewrite the expression as,

$$\begin{gathered} {\log }_2\sqrt[4]{\frac{5x^3}{16y^2{z}^7}}={\log }_2{\left(\frac{5x^3}{16y^2{z}^7}\right)}^{\frac{1}{4}} \\ =\frac{1}{4}{\log }_2\left(\frac{5x^3}{16y^2{z}^7}\right) \end{gathered}$$

Step 3. Further Calculation

Now use property ${\log }_a\frac{M}{N}={\log }_{a}M-{\log }_{a}N$

$$\begin{gathered} \frac{1}{4}{\log }_2\left(\frac{5x^3}{2y^2{z}^4}\right)=\frac{1}{4}\left({\log }_{2}5x^3-{\log }_{2}16y^2{z}^7\right) \\ =\frac{1}{4}{\log }_{2}5x^3-\frac{1}{4}{\log }_{2}16y^2{z}^7 \end{gathered}$$

Now use the property${\log }_a(M·N)={\log }_{a}M+{\log }_{a}N$

$$\begin{gathered} \frac{1}{4}{\log }_2\left(\frac{5x^3}{16y^2{z}^7}\right)=\frac{1}{4}\left({\log }_{2}5+{\log }_2{x}^3\right)-\frac{1}{4}\left({\log }_{2}16+{\log }_2{y}^2+{\log }_2{z}^7\right) \\ =\frac{1}{4}\left({\log }_{2}5+3{\log }_{2}x\right)-\frac{1}{4}\left({\log }_2{2}^4+2{\log }_{2}y+7{\log }_{2}z\right) \\ =\frac{1}{4}\left({\log }_{2}5+3{\log }_{2}x\right)-\frac{1}{4}\left(4+2{\log }_{2}y+7{\log }_{2}z\right) \\ =\frac{1}{4}{\log }_{2}5+\frac{3}{4}{\log }_{2}x-1-\frac{1}{2}{\log }_{2}y-\frac{7}{4}{\log }_{2}z \end{gathered}$$