Precalculus Examples

$\frac{4 \log (36 - x^{2}) - (\log (6 + x) + \log (6 - x))}{(6 + x) (6 - x)}$

Step 1

Simplify $4 \log (36 - x^{2})$ by moving $4$ inside the logarithm.

$\frac{\log ({(36 - x^{2})}^{4}) - (\log (6 + x) + \log (6 - x))}{(6 + x) (6 - x)}$

Step 2

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

$\frac{\log ({(36 - x^{2})}^{4}) - \log ((6 + x) (6 - x))}{(6 + x) (6 - x)}$

Step 3

Expand $(6 + x) (6 - x)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{\log ({(36 - x^{2})}^{4}) - \log (6 (6 - x) + x (6 - x))}{(6 + x) (6 - x)}$

Step 3.2

Apply the distributive property.

$\frac{\log ({(36 - x^{2})}^{4}) - \log (6 \cdot 6 + 6 (- x) + x (6 - x))}{(6 + x) (6 - x)}$

Step 3.3

Apply the distributive property.

$\frac{\log ({(36 - x^{2})}^{4}) - \log (6 \cdot 6 + 6 (- x) + x \cdot 6 + x (- x))}{(6 + x) (6 - x)}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $6$ by $6$ .

$\frac{\log ({(36 - x^{2})}^{4}) - \log (36 + 6 (- x) + x \cdot 6 + x (- x))}{(6 + x) (6 - x)}$

Step 4.1.2

Multiply $- 1$ by $6$ .

$\frac{\log ({(36 - x^{2})}^{4}) - \log (36 - 6 x + x \cdot 6 + x (- x))}{(6 + x) (6 - x)}$

Step 4.1.3

Move $6$ to the left of $x$ .

$\frac{\log ({(36 - x^{2})}^{4}) - \log (36 - 6 x + 6 \cdot x + x (- x))}{(6 + x) (6 - x)}$

Step 4.1.4

Rewrite using the commutative property of multiplication.

$\frac{\log ({(36 - x^{2})}^{4}) - \log (36 - 6 x + 6 x - x \cdot x)}{(6 + x) (6 - x)}$

Step 4.1.5

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

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Step 4.1.5.1

Move $x$ .

$\frac{\log ({(36 - x^{2})}^{4}) - \log (36 - 6 x + 6 x - (x \cdot x))}{(6 + x) (6 - x)}$

Step 4.1.5.2

Multiply $x$ by $x$ .

$\frac{\log ({(36 - x^{2})}^{4}) - \log (36 - 6 x + 6 x - x^{2})}{(6 + x) (6 - x)}$

Step 4.2

Add $- 6 x$ and $6 x$ .

$\frac{\log ({(36 - x^{2})}^{4}) - \log (36 + 0 - x^{2})}{(6 + x) (6 - x)}$

Step 4.3

Add $36$ and $0$ .

$\frac{\log ({(36 - x^{2})}^{4}) - \log (36 - x^{2})}{(6 + x) (6 - x)}$

Step 5

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

$\frac{\log (\frac{{(36 - x^{2})}^{4}}{36 - x^{2}})}{(6 + x) (6 - x)}$

Step 6

Cancel the common factor of ${(36 - x^{2})}^{4}$ and $36 - x^{2}$ .

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Step 6.1

Factor $36 - x^{2}$ out of ${(36 - x^{2})}^{4}$ .

$\frac{\log (\frac{(36 - x^{2}) {(36 - x^{2})}^{3}}{36 - x^{2}})}{(6 + x) (6 - x)}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Multiply by $1$ .

$\frac{\log (\frac{(36 - x^{2}) {(36 - x^{2})}^{3}}{(36 - x^{2}) \cdot 1})}{(6 + x) (6 - x)}$

Step 6.2.2

Cancel the common factor.

$\frac{\log (\frac{(36 - x^{2}) {(36 - x^{2})}^{3}}{(36 - x^{2}) \cdot 1})}{(6 + x) (6 - x)}$

Step 6.2.3

Rewrite the expression.

$\frac{\log (\frac{{(36 - x^{2})}^{3}}{1})}{(6 + x) (6 - x)}$

Step 6.2.4

Divide ${(36 - x^{2})}^{3}$ by $1$ .

$\frac{\log ({(36 - x^{2})}^{3})}{(6 + x) (6 - x)}$