Question

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

$4\log x-2\log y-3\log z$

Short answer

On simplifying the expression we get:

$\log \frac{x^4}{y^2{z}^3}$

Step-by-step solution

Step 1. Given information

The given expression is $4\log x-2\log y-3\log z$.

We have to condense the expression using logarithm properties.

Step 2. Identify the logarithm property

Use the power property of logarithm:

$p{\log }_a\left(M\right)={\log }_a{\left(M\right)}^p$

Step 3. Use the power property of logarithm

$$\begin{gathered} 4\log x-2\log y-3\log z \\ =\log x^4-\log y^2-\log z^3 \end{gathered}$$

Step 4. Apply the quotient property of logarithm

Quotient property of logarithm is ${\log }_{a}M-{\log }_{a}N={\log }_a\left(\frac{M}{N}\right)$

$$\begin{gathered} \log x^4-\log y^2-\log z^3 \\ =\log \left(\frac{x^4}{y^2{z}^3}\right) \end{gathered}$$