Precalculus Examples

$\log (x^{2} + 16) - \log (x + 4) = 1 + \log (x - 4)$

Step 1

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

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

Step 2

Move all the terms containing a logarithm to the left side of the equation.

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

Step 3

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Multiply $\frac{x^{2} + 16}{x + 4}$ by $\frac{1}{x - 4}$ .

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

Step 6

Rewrite $\log (\frac{x^{2} + 16}{(x + 4) (x - 4)}) = 1$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ $\neq$ $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{1} = \frac{x^{2} + 16}{(x + 4) (x - 4)}$

Step 7

Cross multiply to remove the fraction.

$x^{2} + 16 = 10 ((x + 4) (x - 4))$

Step 8

Simplify $10 ((x + 4) (x - 4))$ .

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Step 8.1

Expand $(x + 4) (x - 4)$ using the FOIL Method.

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Step 8.1.1

Apply the distributive property.

$x^{2} + 16 = 10 (x (x - 4) + 4 (x - 4))$

Step 8.1.2

Apply the distributive property.

$x^{2} + 16 = 10 (x \cdot x + x \cdot - 4 + 4 (x - 4))$

Step 8.1.3

Apply the distributive property.

$x^{2} + 16 = 10 (x \cdot x + x \cdot - 4 + 4 x + 4 \cdot - 4)$

Step 8.2

Simplify terms.

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Step 8.2.1

Combine the opposite terms in $x \cdot x + x \cdot - 4 + 4 x + 4 \cdot - 4$ .

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Step 8.2.1.1

Reorder the factors in the terms $x \cdot - 4$ and $4 x$ .

$x^{2} + 16 = 10 (x \cdot x - 4 x + 4 x + 4 \cdot - 4)$

Step 8.2.1.2

Add $- 4 x$ and $4 x$ .

$x^{2} + 16 = 10 (x \cdot x + 0 + 4 \cdot - 4)$

Step 8.2.1.3

Add $x \cdot x$ and $0$ .

$x^{2} + 16 = 10 (x \cdot x + 4 \cdot - 4)$

Step 8.2.2

Simplify each term.

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Step 8.2.2.1

Multiply $x$ by $x$ .

$x^{2} + 16 = 10 (x^{2} + 4 \cdot - 4)$

Step 8.2.2.2

Multiply $4$ by $- 4$ .

$x^{2} + 16 = 10 (x^{2} - 16)$

Step 8.2.3

Simplify by multiplying through.

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Step 8.2.3.1

Apply the distributive property.

$x^{2} + 16 = 10 x^{2} + 10 \cdot - 16$

Step 8.2.3.2

Multiply $10$ by $- 16$ .

$x^{2} + 16 = 10 x^{2} - 160$

Step 9

Move all terms containing $x$ to the left side of the equation.

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Step 9.1

Subtract $10 x^{2}$ from both sides of the equation.

$x^{2} + 16 - 10 x^{2} = - 160$

Step 9.2

Subtract $10 x^{2}$ from $x^{2}$ .

$- 9 x^{2} + 16 = - 160$

Step 10

Factor the left side of the equation.

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Step 10.1

Rewrite $16$ as $4^{2}$ .

$- 9 x^{2} + 4^{2} = - 160$

Step 10.2

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$- {(3 x)}^{2} + 4^{2} = - 160$

Step 10.3

Reorder $- {(3 x)}^{2}$ and $4^{2}$ .

$4^{2} - {(3 x)}^{2} = - 160$

Step 10.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = 3 x$ .

$(4 + 3 x) (4 - (3 x)) = - 160$

Step 10.5

Multiply $3$ by $- 1$ .

$(4 + 3 x) (4 - 3 x) = - 160$

Step 11

Simplify $(4 + 3 x) (4 - 3 x)$ .

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Step 11.1

Expand $(4 + 3 x) (4 - 3 x)$ using the FOIL Method.

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Step 11.1.1

Apply the distributive property.

$4 (4 - 3 x) + 3 x (4 - 3 x) = - 160$

Step 11.1.2

Apply the distributive property.

$4 \cdot 4 + 4 (- 3 x) + 3 x (4 - 3 x) = - 160$

Step 11.1.3

Apply the distributive property.

$4 \cdot 4 + 4 (- 3 x) + 3 x \cdot 4 + 3 x (- 3 x) = - 160$

Step 11.2

Simplify terms.

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Step 11.2.1

Combine the opposite terms in $4 \cdot 4 + 4 (- 3 x) + 3 x \cdot 4 + 3 x (- 3 x)$ .

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Step 11.2.1.1

Reorder the factors in the terms $4 (- 3 x)$ and $3 x \cdot 4$ .

$4 \cdot 4 - 3 \cdot 4 x + 3 \cdot 4 x + 3 x (- 3 x) = - 160$

Step 11.2.1.2

Add $- 3 \cdot 4 x$ and $3 \cdot 4 x$ .

$4 \cdot 4 + 0 + 3 x (- 3 x) = - 160$

Step 11.2.1.3

Add $4 \cdot 4$ and $0$ .

$4 \cdot 4 + 3 x (- 3 x) = - 160$

Step 11.2.2

Simplify each term.

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Step 11.2.2.1

Multiply $4$ by $4$ .

$16 + 3 x (- 3 x) = - 160$

Step 11.2.2.2

Rewrite using the commutative property of multiplication.

$16 + 3 \cdot - 3 x \cdot x = - 160$

Step 11.2.2.3

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

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Step 11.2.2.3.1

Move $x$ .

$16 + 3 \cdot - 3 (x \cdot x) = - 160$

Step 11.2.2.3.2

Multiply $x$ by $x$ .

$16 + 3 \cdot - 3 x^{2} = - 160$

Step 11.2.2.4

Multiply $3$ by $- 3$ .

$16 - 9 x^{2} = - 160$

Step 12

Move all terms not containing $x$ to the right side of the equation.

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Step 12.1

Subtract $16$ from both sides of the equation.

$- 9 x^{2} = - 160 - 16$

Step 12.2

Subtract $16$ from $- 160$ .

$- 9 x^{2} = - 176$

Step 13

Divide each term in $- 9 x^{2} = - 176$ by $- 9$ and simplify.

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Step 13.1

Divide each term in $- 9 x^{2} = - 176$ by $- 9$ .

$\frac{- 9 x^{2}}{- 9} = \frac{- 176}{- 9}$

Step 13.2

Simplify the left side.

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Step 13.2.1

Cancel the common factor of $- 9$ .

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Step 13.2.1.1

Cancel the common factor.

$\frac{- 9 x^{2}}{- 9} = \frac{- 176}{- 9}$

Step 13.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 176}{- 9}$

Step 13.3

Simplify the right side.

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Step 13.3.1

Dividing two negative values results in a positive value.

$x^{2} = \frac{176}{9}$

Step 14

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{176}{9}}$

Step 15

Simplify $\pm \sqrt{\frac{176}{9}}$ .

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Step 15.1

Rewrite $\sqrt{\frac{176}{9}}$ as $\frac{\sqrt{176}}{\sqrt{9}}$ .

$x = \pm \frac{\sqrt{176}}{\sqrt{9}}$

Step 15.2

Simplify the numerator.

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Step 15.2.1

Rewrite $176$ as $4^{2} \cdot 11$ .

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Step 15.2.1.1

Factor $16$ out of $176$ .

$x = \pm \frac{\sqrt{16 (11)}}{\sqrt{9}}$

Step 15.2.1.2

Rewrite $16$ as $4^{2}$ .

$x = \pm \frac{\sqrt{4^{2} \cdot 11}}{\sqrt{9}}$

Step 15.2.2

Pull terms out from under the radical.

$x = \pm \frac{4 \sqrt{11}}{\sqrt{9}}$

Step 15.3

Simplify the denominator.

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Step 15.3.1

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

$x = \pm \frac{4 \sqrt{11}}{\sqrt{3^{2}}}$

Step 15.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{4 \sqrt{11}}{3}$

Step 16

The complete solution is the result of both the positive and negative portions of the solution.

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Step 16.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{4 \sqrt{11}}{3}$

Step 16.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{4 \sqrt{11}}{3}$

Step 16.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{4 \sqrt{11}}{3}, - \frac{4 \sqrt{11}}{3}$

Step 17

Exclude the solutions that do not make $\log (x^{2} + 16) - \log (x + 4) = 1 + \log (x - 4)$ true.

$x = \frac{4 \sqrt{11}}{3}$

Step 18

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4 \sqrt{11}}{3}$

Decimal Form:

$x = 4.42216638 \dots$