Precalculus Examples

$\log_{3} (x - 5) + \log_{3} (x + 19)$

Step 1

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

$\log_{3} ((x - 5) (x + 19))$

Step 2

Expand $(x - 5) (x + 19)$ using the FOIL Method.

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Apply the distributive property.

$\log_{3} (x (x + 19) - 5 (x + 19))$

Apply the distributive property.

$\log_{3} (x \cdot x + x \cdot 19 - 5 (x + 19))$

Apply the distributive property.

$\log_{3} (x \cdot x + x \cdot 19 - 5 x - 5 \cdot 19)$

Step 3

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ .

$\log_{3} (x^{2} + x \cdot 19 - 5 x - 5 \cdot 19)$

Move $19$ to the left of $x$ .

$\log_{3} (x^{2} + 19 \cdot x - 5 x - 5 \cdot 19)$

Multiply $- 5$ by $19$ .

$\log_{3} (x^{2} + 19 x - 5 x - 95)$

Subtract $5 x$ from $19 x$ .

$\log_{3} (x^{2} + 14 x - 95)$

Step 4

Rewrite $\log_{3} (x^{2} + 14 x - 95)$ using the change of base formula.

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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 $0$ .

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

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

$\frac{\log (x^{2} + 14 x - 95)}{\log (3)}$