Question

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

$\log \left(x^2+2x+1\right)-2\log \left(x+1\right)$

Short answer

On simplifying the expression we get: $0$

Step-by-step solution

Step 1. Given information

The given expression is$\log \left(x^2+2x+1\right)-2\log \left(x+1\right)$

We have to condense the expression using logarithm properties.

Step 2. Identify the logarithm property

Use the power property of logarithm:

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

Step 3. Use the power property of logarithm

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

Since,

${(a+b)}^2=a^2+2ab+b^2$

Therefore,

$$\begin{gathered} \log \left(x^2+2x+1\right)-\log {\left(x+1\right)}^2 \\ =\log {\left(x+1\right)}^2-\log {\left(x+1\right)}^2 \\ =\log 1 \\ =0 \end{gathered}$$