Precalculus Examples

log (x) + log (x^{2} - 16) - log (11) - log (x + 4)

$\log (x) + \log (x^{2} - 16) - \log (11) - \log (x + 4)$

Step 1

Use the product property of logarithms,

{log}_{b} (x) + {log}_{b} (y) = {log}_{b} (x y)

$\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

log (x (x^{2} - 16)) - log (11) - log (x + 4)

$\log (x (x^{2} - 16)) - \log (11) - \log (x + 4)$

Step 2

Use the quotient property of logarithms,

{log}_{b} (x) - {log}_{b} (y) = {log}_{b} (\frac{x}{y})

$\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

log (\frac{x (x^{2} - 16)}{11}) - log (x + 4)

$\log (\frac{x (x^{2} - 16)}{11}) - \log (x + 4)$

Step 3

Use the quotient property of logarithms,

{log}_{b} (x) - {log}_{b} (y) = {log}_{b} (\frac{x}{y})

$\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{\frac{x (x^{2} - 16)}{11}}{x + 4})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{x (x^{2} - 16)}{11} \cdot \frac{1}{x + 4})$

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Rewrite

$16$ as

$4^{2}$ .

$\log (\frac{x (x^{2} - 4^{2})}{11} \cdot \frac{1}{x + 4})$

Step 5.2

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

$b = 4$ .

$\log (\frac{x (x + 4) (x - 4)}{11} \cdot \frac{1}{x + 4})$

Step 6

Cancel the common factor of

$x + 4$ .

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Step 6.1

Factor

$x + 4$ out of

$x (x + 4) (x - 4)$ .

$\log (\frac{(x + 4) (x (x - 4))}{11} \cdot \frac{1}{x + 4})$

Step 6.2

Cancel the common factor.

$\log (\frac{(x + 4) (x (x - 4))}{11} \cdot \frac{1}{x + 4})$

Step 6.3

Rewrite the expression.

$\log (\frac{x (x - 4)}{11})$

$\log (x) + \log (x^{2} - 16) - \log (11) - \log (x + 4)$

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