Precalculus Examples

$\log (x^{2} + 3 x + 2) - 2 \log (x + 1)$

Step 1

Simplify $- 2 \log (x + 1)$ by moving $2$ inside the logarithm.

$\log (x^{2} + 3 x + 2) - \log ({(x + 1)}^{2})$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{x^{2} + 3 x + 2}{{(x + 1)}^{2}})$

Step 3

Factor $x^{2} + 3 x + 2$ using the AC method.

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Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $3$ .

$1, 2$

Step 3.2

Write the factored form using these integers.

$\log (\frac{(x + 1) (x + 2)}{{(x + 1)}^{2}})$

Step 4

Cancel the common factors.

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Step 4.1

Factor $x + 1$ out of ${(x + 1)}^{2}$ .

$\log (\frac{(x + 1) (x + 2)}{(x + 1) (x + 1)})$

Step 4.2

Cancel the common factor.

$\log (\frac{(x + 1) (x + 2)}{(x + 1) (x + 1)})$

Step 4.3

Rewrite the expression.

$\log (\frac{x + 2}{x + 1})$