Question

Use the Properties of Logarithms to condense the logarithm $2\log x+2\log (x+1).$Simplify, if possible.

Short answer

The condensed form of the logarithm is$\log x^2{\left(x+1\right)}^2.$

Step-by-step solution

Step 1. Given Infomation

The given logarithm is $2\log x+2\log \left(x+1\right).$

We have to use the Properties of Logarithms to condense the logarithm.

Step 2. Use the Power Property of the logarithm

The power property of the logarithm is ${\log }_a{M}^p=p{\log }_{a}M$

So,

role="math" localid="1644996806106" $$\begin{gathered} 2\log x+2\log \left(x+1\right) \\ =\log x^2+\log {\left(x+1\right)}^2 \end{gathered}$$

Step 3. Use the Product Property of logarithm

The Product Property of logarithm is ${\log }_{a}M·N={\log }_{a}M+{\log }_{a}N.$

So,

$$\begin{gathered} =\log x^2+\log {\left(x+1\right)}^2 \\ =\log x^2{\left(x+1\right)}^2 \end{gathered}$$