Precalculus Examples

$\log_{4} (x^{2} - 7 x + 10) - \log_{4} (x - 2) + \log_{4} (x + 2)$

Step 1

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

$\log_{4} (\frac{x^{2} - 7 x + 10}{x - 2}) + \log_{4} (x + 2)$

Step 2

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

$\log_{4} (\frac{x^{2} - 7 x + 10}{x - 2} (x + 2))$

Step 3

Factor $x^{2} - 7 x + 10$ 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 $10$ and whose sum is $- 7$ .

$- 5, - 2$

Step 3.2

Write the factored form using these integers.

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

Step 4

Cancel the common factor of $x - 2$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Divide $x - 5$ by $1$ .

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

Step 5

Expand $(x - 5) (x + 2)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $x$ by $x$ .

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

Step 6.1.2

Move $2$ to the left of $x$ .

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

Step 6.1.3

Multiply $- 5$ by $2$ .

$\log_{4} (x^{2} + 2 x - 5 x - 10)$

Step 6.2

Subtract $5 x$ from $2 x$ .

$\log_{4} (x^{2} - 3 x - 10)$