Precalculus Examples

$4 \log (49 - x^{2}) - (\log (7 + x) + \log (7 - x))$

Step 1

Simplify each term.

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Step 1.1

Simplify $4 \log (49 - x^{2})$ by moving $4$ inside the logarithm.

$\log ({(49 - x^{2})}^{4}) - (\log (7 + x) + \log (7 - x))$

Step 1.2

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

$\log ({(49 - x^{2})}^{4}) - \log ((7 + x) (7 - x))$

Step 1.3

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

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Step 1.3.1

Apply the distributive property.

$\log ({(49 - x^{2})}^{4}) - \log (7 (7 - x) + x (7 - x))$

Step 1.3.2

Apply the distributive property.

$\log ({(49 - x^{2})}^{4}) - \log (7 \cdot 7 + 7 (- x) + x (7 - x))$

Step 1.3.3

Apply the distributive property.

$\log ({(49 - x^{2})}^{4}) - \log (7 \cdot 7 + 7 (- x) + x \cdot 7 + x (- x))$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $7$ by $7$ .

$\log ({(49 - x^{2})}^{4}) - \log (49 + 7 (- x) + x \cdot 7 + x (- x))$

Step 1.4.1.2

Multiply $- 1$ by $7$ .

$\log ({(49 - x^{2})}^{4}) - \log (49 - 7 x + x \cdot 7 + x (- x))$

Step 1.4.1.3

Move $7$ to the left of $x$ .

$\log ({(49 - x^{2})}^{4}) - \log (49 - 7 x + 7 \cdot x + x (- x))$

Step 1.4.1.4

Rewrite using the commutative property of multiplication.

$\log ({(49 - x^{2})}^{4}) - \log (49 - 7 x + 7 x - x \cdot x)$

Step 1.4.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 1.4.1.5.1

Move $x$ .

$\log ({(49 - x^{2})}^{4}) - \log (49 - 7 x + 7 x - (x \cdot x))$

Step 1.4.1.5.2

Multiply $x$ by $x$ .

$\log ({(49 - x^{2})}^{4}) - \log (49 - 7 x + 7 x - x^{2})$

Step 1.4.2

Add $- 7 x$ and $7 x$ .

$\log ({(49 - x^{2})}^{4}) - \log (49 + 0 - x^{2})$

Step 1.4.3

Add $49$ and $0$ .

$\log ({(49 - x^{2})}^{4}) - \log (49 - x^{2})$

Step 2

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

$\log (\frac{{(49 - x^{2})}^{4}}{49 - x^{2}})$

Step 3

Cancel the common factor of ${(49 - x^{2})}^{4}$ and $49 - x^{2}$ .

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Step 3.1

Factor $49 - x^{2}$ out of ${(49 - x^{2})}^{4}$ .

$\log (\frac{(49 - x^{2}) {(49 - x^{2})}^{3}}{49 - x^{2}})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Multiply by $1$ .

$\log (\frac{(49 - x^{2}) {(49 - x^{2})}^{3}}{(49 - x^{2}) \cdot 1})$

Step 3.2.2

Cancel the common factor.

$\log (\frac{(49 - x^{2}) {(49 - x^{2})}^{3}}{(49 - x^{2}) \cdot 1})$

Step 3.2.3

Rewrite the expression.

$\log (\frac{{(49 - x^{2})}^{3}}{1})$

Step 3.2.4

Divide ${(49 - x^{2})}^{3}$ by $1$ .

$\log ({(49 - x^{2})}^{3})$