Question

Write each expression as a single logarithm.

$\log \left(x^2+3x+2\right)-2\log \left(x+1\right)$

Short answer

The result is$\begin{array}{rcl}& & \log \frac{\left(x+2\right)}{\left(x+1\right)}\end{array}$.

Step-by-step solution

Step 1. Given information.

Consider the given information.

$\log \left(x^2+3x+2\right)-2\log \left(x+1\right)$

Step 2. Calculation.

Apply the property ${\log }_a\left(\frac{M}{N}\right)={\log }_{a}M-{\log }_{a}N$.

$\begin{array}{rcl}\log \left(x^2+3x+2\right)-2\log \left(x+1\right)& =& \log \left(x^2+2x+x+2\right)-\log {\left(x+1\right)}^2\\ & =& \log \frac{\left(\left(x+2\right)\left(x+1\right)\right)}{{\left(x+1\right)}^2}\\ & =& \log \frac{\left(x+2\right)}{\left(x+1\right)}\\ & & \end{array}$

Step 2. Calculation.

The obtained result is$\begin{array}{rcl}& & \log \frac{\left(x+2\right)}{\left(x+1\right)}\end{array}$.