Question

In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

$\ln \frac{\sqrt{3x^2-y^2}}{z^4}$

Short answer

logarithm expression as a sum of logarithms will be$\ln \frac{\sqrt{3x^2-y^2}}{z^4}=\frac{1}{2}\ln 3+\ln x-\ln y-4\ln z$

Step-by-step solution

Step 1. Given data

The given logarithm expression is$\ln \frac{\sqrt{3x^2-y^2}}{z^4}$

Step 2. Apply the quotient properties of the logarithm

Use quotient properties of the logarithm $\ln \left(\frac{m}{n}\right)=\ln \left(m\right)-\ln \left(n\right)$

$$\begin{gathered} \ln \frac{\sqrt{3x^2-y^2}}{z^4}=\ln \sqrt{3x^2-y^2}-\ln z^4 \\ \ln \frac{\sqrt{3x^2-y^2}}{z^4}=\ln {\left(3x^2-y^2\right)}^{\frac{1}{2}}-\ln z^4 \end{gathered}$$

Step 3. Apply the power properties of the logarithm

Use power properties of the logarithm $\ln \left(m^p\right)=p\ln \left(m\right)$

$$\begin{gathered} \ln \frac{\sqrt{3x^2-y^2}}{z^4}=\ln {\left(3x^2-y^2\right)}^{\frac{1}{2}}-\ln z^4 \\ \ln \frac{\sqrt{3x^2-y^2}}{z^4}=\frac{1}{2}\ln \left(3x^2-y^2\right)-4\ln z \end{gathered}$$

Step 4. Apply the quotient properties of the logarithm

Use quotient properties of the logarithm $\ln \left(\frac{m}{n}\right)=\ln \left(m\right)-\ln \left(n\right)$

role="math" localid="1645023256402" $$\begin{gathered} \ln \frac{\sqrt{3x^2-y^2}}{z^4}=\frac{1}{2}\ln \left(3x^2-y^2\right)-4\ln z \\ \ln \frac{\sqrt{3x^2-y^2}}{z^4}=\frac{1}{2}\ln 3x^2-\frac{1}{2}\ln y^2-4\ln z \end{gathered}$$

Step 5. Apply the product properties of logarithm

Use product property of logarithm$\ln \left(m·n\right)=\ln \left(m\right)+\ln \left(n\right)$

role="math" localid="1645023403775" $$\begin{gathered} \ln \frac{\sqrt{3x^2-y^2}}{z^4}=\frac{1}{2}\ln 3x^2-\frac{1}{2}\ln y^2-4\ln z \\ \ln \frac{\sqrt{3x^2-y^2}}{z^4}=\frac{1}{2}\ln 3+\frac{1}{2}\ln x^2-\frac{1}{2}\ln y^2-4\ln z \end{gathered}$$

Step 6. Apply the power properties of the logarithm

Use power properties of the logarithm $\ln \left(m^p\right)=p\ln \left(m\right)$

role="math" localid="1645023480021" $$\begin{gathered} \ln \frac{\sqrt{3x^2-y^2}}{z^4}=\frac{1}{2}\ln 3+\frac{1}{2}\ln x^2-\frac{1}{2}\ln y^2-4\ln z \\ \ln \frac{\sqrt{3x^2-y^2}}{z^4}=\frac{1}{2}\ln 3+\frac{2}{2}\ln x-\frac{2}{2}\ln y-4\ln z \\ \ln \frac{\sqrt{3x^2-y^2}}{z^4}=\frac{1}{2}\ln 3+\ln x-\ln y-4\ln z \end{gathered}$$