Question

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

${\log }_5\sqrt[3]{\frac{3x^2}{4y^{3}z}}$

Short answer

The simplified expression is,

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

Step-by-step solution

Step 1. Given and explanation  

We are given an expression,

${\log }_5\sqrt[3]{\frac{3x^2}{4y^{3}z}}$

Use the properties of logarithms to simplify the given expression.

Step 2. Simplifying the expression  

The given expression can be written as,

${\log }_5\sqrt[3]{\frac{3x^2}{4y^{3}z}}={\log }_5{\left(\frac{3x^2}{4y^{3}z}\right)}^{\frac{1}{3}}$

using power property of logarithm , we get

${\log }_5\sqrt[3]{\frac{3x^2}{4y^{3}z}}=\frac{1}{3}{\log }_5\left(\frac{3x^2}{4y^{3}z}\right)$

Using quotient property of logarithm , we get

${\log }_5\sqrt[3]{\frac{3x^2}{4y^{3}z}}=\frac{1}{3}\left({\log }_{5}3x^2-{\log }_{5}4y^{3}z\right)$

Using product property of logarithm , we get

${\log }_5\sqrt[3]{\frac{3x^2}{4y^{3}z}}=\frac{1}{3}\left({\log }_{5}3+{\log }_5{x}^2\right)-\frac{1}{3}\left({\log }_{5}4+{\log }_5{y}^3+{\log }_{5}z\right)$

Step 3. Simplifying

Now, using power property of logarithm , we get

Using , we get

$\begin{array}{l}{\log }_5\sqrt[3]{\frac{3x^2}{4y^{3}z}}=\frac{1}{3}\left({\log }_{5}3+2{\log }_{5}x\right)-\frac{1}{3}\left({\log }_{5}4+3{\log }_{5}y+{\log }_{5}z\right)\\ {\log }_5\sqrt[3]{\frac{3x^2}{4y^{3}z}}=\frac{1}{3}{\log }_{5}3+\frac{2}{3}{\log }_{5}x-\frac{1}{3}{\log }_{5}4-{\log }_{5}y-\frac{1}{3}{\log }_{5}z\end{array}$