Calculus Examples

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

Step 1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 2

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

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

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 $b = 1$ .

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x - 1$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Divide $5 (x + 1)$ by $1$ .

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Multiply $5$ by $1$ .

$\log_{2} (5 x + 5)$