Precalculus Examples

$\log_{4} (3 x - 7) + \log_{4} (x - 3)$

Step 1

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

$\log_{4} ((3 x - 7) (x - 3))$

Step 2

Expand $(3 x - 7) (x - 3)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\log_{4} (3 x (x - 3) - 7 (x - 3))$

Step 2.2

Apply the distributive property.

$\log_{4} (3 x \cdot x + 3 x \cdot - 3 - 7 (x - 3))$

Step 2.3

Apply the distributive property.

$\log_{4} (3 x \cdot x + 3 x \cdot - 3 - 7 x - 7 \cdot - 3)$

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 $x$ by adding the exponents.

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Step 3.1.1.1

Move $x$ .

$\log_{4} (3 (x \cdot x) + 3 x \cdot - 3 - 7 x - 7 \cdot - 3)$

Step 3.1.1.2

Multiply $x$ by $x$ .

$\log_{4} (3 x^{2} + 3 x \cdot - 3 - 7 x - 7 \cdot - 3)$

Step 3.1.2

Multiply $- 3$ by $3$ .

$\log_{4} (3 x^{2} - 9 x - 7 x - 7 \cdot - 3)$

Step 3.1.3

Multiply $- 7$ by $- 3$ .

$\log_{4} (3 x^{2} - 9 x - 7 x + 21)$

Step 3.2

Subtract $7 x$ from $- 9 x$ .

$\log_{4} (3 x^{2} - 16 x + 21)$

Step 4

Rewrite $\log_{4} (3 x^{2} - 16 x + 21)$ 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 $b = 10$ .

$\frac{\log (3 x^{2} - 16 x + 21)}{\log (4)}$