Precalculus Examples

$2 \log_{4} (2 x) + 0.5 \log_{4} (x + 3) - \log_{4} (x^{2} - 9)$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{4} (2 x)$ by moving $2$ inside the logarithm.

$\log_{4} ({(2 x)}^{2}) + 0.5 \log_{4} (x + 3) - \log_{4} (x^{2} - 9)$

Step 1.2

Apply the product rule to $2 x$ .

$\log_{4} (2^{2} x^{2}) + 0.5 \log_{4} (x + 3) - \log_{4} (x^{2} - 9)$

Step 1.3

Raise $2$ to the power of $2$ .

$\log_{4} (4 x^{2}) + 0.5 \log_{4} (x + 3) - \log_{4} (x^{2} - 9)$

Step 1.4

Simplify $0.5 \log_{4} (x + 3)$ by moving $0.5$ inside the logarithm.

$\log_{4} (4 x^{2}) + \log_{4} ({(x + 3)}^{0.5}) - \log_{4} (x^{2} - 9)$

Step 2

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

$\log_{4} (4 x^{2} {(x + 3)}^{0.5}) - \log_{4} (x^{2} - 9)$

Step 3

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

$\log_{4} (\frac{4 x^{2} {(x + 3)}^{0.5}}{x^{2} - 9})$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $9$ as $3^{2}$ .

$\log_{4} (\frac{4 x^{2} {(x + 3)}^{0.5}}{x^{2} - 3^{2}})$

Step 4.2

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 $b = 3$ .

$\log_{4} (\frac{4 x^{2} {(x + 3)}^{0.5}}{(x + 3) (x - 3)})$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Cancel the common factor of ${(x + 3)}^{0.5}$ and $x + 3$ .

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Step 5.1.1

Factor $x + 3$ out of $4 x^{2} {(x + 3)}^{0.5}$ .

$\log_{4} (\frac{(x + 3) (4 x^{2} {(x + 3)}^{- 0.5})}{(x + 3) (x - 3)})$

Step 5.1.2

Cancel the common factors.

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Step 5.1.2.1

Cancel the common factor.

$\log_{4} (\frac{(x + 3) (4 x^{2} {(x + 3)}^{- 0.5})}{(x + 3) (x - 3)})$

Step 5.1.2.2

Rewrite the expression.

$\log_{4} (\frac{4 x^{2} {(x + 3)}^{- 0.5}}{x - 3})$

Step 5.2

Simplify the expression.

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Step 5.2.1

Move ${(x + 3)}^{- 0.5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\log_{4} (\frac{4 x^{2}}{(x - 3) {(x + 3)}^{0.5}})$

Step 5.2.2

Reorder factors in $\log_{4} (\frac{4 x^{2}}{(x - 3) {(x + 3)}^{0.5}})$ .

$\log_{4} (\frac{4 x^{2}}{{(x + 3)}^{0.5} (x - 3)})$