Precalculus Examples

$\log_{5} (\frac{x^{2} + 2 x + 1}{x^{2}})$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

$\log_{5} (\frac{x^{2} + 2 x + 1^{2}}{x^{2}})$

Step 1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 1.3

Rewrite the polynomial.

$\log_{5} (\frac{x^{2} + 2 \cdot x \cdot 1 + 1^{2}}{x^{2}})$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

$\log_{5} (\frac{{(x + 1)}^{2}}{x^{2}})$

Step 2

Rewrite $\log_{5} (\frac{{(x + 1)}^{2}}{x^{2}})$ using the change of base formula.

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Step 2.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 2.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (\frac{{(x + 1)}^{2}}{x^{2}})}{\log (5)}$