Question

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

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

Short answer

The simplified expression is,

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

Step-by-step solution

Step 1. Given and explanation  

We are given an expression,

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

Use the properties of logarithms to simplify the given expression.

Step 2. Simplifying the expression  

The expression can be written as,

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

Using power property of logarithm , we get

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

Using quotient property of logarithm , we get

role="math" localid="1645015725642" $$\begin{gathered} {\log }_2\sqrt[4]{\frac{5x^3}{2y^2{z}^4}}=\frac{1}{4}\left({\log }_{2}5x^3-{\log }_{2}2y^2{z}^4\right) \\ {\log }_2\sqrt[4]{\frac{5x^3}{2y^2{z}^4}}=\frac{1}{4}{\log }_{2}5x^3-\frac{1}{4}{\log }_{2}2y^2{z}^4 \end{gathered}$$

Step 3. Simplifying

Now, using product property of logarithm , we get

${\log }_2\sqrt[4]{\frac{5x^3}{2y^2{z}^4}}=\frac{1}{4}\left({\log }_{2}5+{\log }_2{x}^3\right)-\frac{1}{4}\left({\log }_{2}2+{\log }_2{y}^2+{\log }_2{z}^4\right)$

using power property of logarithm , we get

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