Precalculus Examples

$\log_{4} (x^{2} - 1) - 5 \log_{4} (x + 1)$

Step 1

Simplify $- 5 \log_{4} (x + 1)$ by moving $5$ inside the logarithm.

$\log_{4} (x^{2} - 1) - \log_{4} ({(x + 1)}^{5})$

Step 2

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

$\log_{4} (\frac{x^{2} - 1}{{(x + 1)}^{5}})$

Step 3

Simplify the numerator.

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Step 3.1

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

$\log_{4} (\frac{x^{2} - 1^{2}}{{(x + 1)}^{5}})$

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\log_{4} (\frac{(x + 1) (x - 1)}{{(x + 1)}^{5}})$

Step 4

Cancel the common factors.

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Step 4.1

Factor $x + 1$ out of ${(x + 1)}^{5}$ .

$\log_{4} (\frac{(x + 1) (x - 1)}{(x + 1) {(x + 1)}^{4}})$

Step 4.2

Cancel the common factor.

$\log_{4} (\frac{(x + 1) (x - 1)}{(x + 1) {(x + 1)}^{4}})$

Step 4.3

Rewrite the expression.

$\log_{4} (\frac{x - 1}{{(x + 1)}^{4}})$

Step 5

Rewrite $\log_{4} (\frac{x - 1}{{(x + 1)}^{4}})$ using the change of base formula.

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Step 5.1

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 5.2

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

$\frac{\log (\frac{x - 1}{{(x + 1)}^{4}})}{\log (4)}$