Precalculus Examples

$\log_{16} (x - 4) + \log_{16} (x + 2)$

Step 1

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

$\log_{16} ((x - 4) (x + 2))$

Step 2

Expand $(x - 4) (x + 2)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\log_{16} (x (x + 2) - 4 (x + 2))$

Step 2.2

Apply the distributive property.

$\log_{16} (x \cdot x + x \cdot 2 - 4 (x + 2))$

Step 2.3

Apply the distributive property.

$\log_{16} (x \cdot x + x \cdot 2 - 4 x - 4 \cdot 2)$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

$\log_{16} (x^{2} + x \cdot 2 - 4 x - 4 \cdot 2)$

Step 3.1.2

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

$\log_{16} (x^{2} + 2 \cdot x - 4 x - 4 \cdot 2)$

Step 3.1.3

Multiply $- 4$ by $2$ .

$\log_{16} (x^{2} + 2 x - 4 x - 8)$

Step 3.2

Subtract $4 x$ from $2 x$ .

$\log_{16} (x^{2} - 2 x - 8)$

Step 4

Rewrite $\log_{16} (x^{2} - 2 x - 8)$ using the change of base formula.

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Step 4.1

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 4.2

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

$\frac{\log (x^{2} - 2 x - 8)}{\log (16)}$