Precalculus Examples

$2 \log_{4} (x + 4) - \log_{4} (x + 12) = 1$

Step 1

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

$2 \log_{4} (x + 4) - \log_{4} (x + 12) = 1$

Step 2

Simplify the left side.

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Step 2.1

Simplify $2 \log_{4} (x + 4) - \log_{4} (x + 12)$ .

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Step 2.1.1

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

$\log_{4} ({(x + 4)}^{2}) - \log_{4} (x + 12) = 1$

Step 2.1.2

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

$\log_{4} (\frac{{(x + 4)}^{2}}{x + 12}) = 1$

Step 3

Rewrite $\log_{4} (\frac{{(x + 4)}^{2}}{x + 12}) = 1$ 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 $b^{y} = x$ .

$4^{1} = \frac{{(x + 4)}^{2}}{x + 12}$

Step 4

Cross multiply to remove the fraction.

${(x + 4)}^{2} = 4 (x + 12)$

Step 5

Simplify $4 (x + 12)$ .

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Step 5.1

Apply the distributive property.

${(x + 4)}^{2} = 4 x + 4 \cdot 12$

Step 5.2

Multiply $4$ by $12$ .

${(x + 4)}^{2} = 4 x + 48$

Step 6

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

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Step 6.1

Subtract $4 x$ from both sides of the equation.

${(x + 4)}^{2} - 4 x = 48$

Step 6.2

Simplify each term.

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Step 6.2.1

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$(x + 4) (x + 4) - 4 x = 48$

Step 6.2.2

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

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Step 6.2.2.1

Apply the distributive property.

$x (x + 4) + 4 (x + 4) - 4 x = 48$

Step 6.2.2.2

Apply the distributive property.

$x \cdot x + x \cdot 4 + 4 (x + 4) - 4 x = 48$

Step 6.2.2.3

Apply the distributive property.

$x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4 - 4 x = 48$

Step 6.2.3

Simplify and combine like terms.

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Step 6.2.3.1

Simplify each term.

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Step 6.2.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 4 + 4 x + 4 \cdot 4 - 4 x = 48$

Step 6.2.3.1.2

Move $4$ to the left of $x$ .

$x^{2} + 4 \cdot x + 4 x + 4 \cdot 4 - 4 x = 48$

Step 6.2.3.1.3

Multiply $4$ by $4$ .

$x^{2} + 4 x + 4 x + 16 - 4 x = 48$

Step 6.2.3.2

Add $4 x$ and $4 x$ .

$x^{2} + 8 x + 16 - 4 x = 48$

Step 6.3

Subtract $4 x$ from $8 x$ .

$x^{2} + 4 x + 16 = 48$

Step 7

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

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Step 7.1

Subtract $16$ from both sides of the equation.

$x^{2} + 4 x = 48 - 16$

Step 7.2

Subtract $16$ from $48$ .

$x^{2} + 4 x = 32$

Step 8

Subtract $32$ from both sides of the equation.

$x^{2} + 4 x - 32 = 0$

Step 9

Factor $x^{2} + 4 x - 32$ using the AC method.

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Step 9.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 32$ and whose sum is $4$ .

$- 4, 8$

Step 9.2

Write the factored form using these integers.

$(x - 4) (x + 8) = 0$

Step 10

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 4 = 0$

$x + 8 = 0$

Step 11

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 11.2

Add $4$ to both sides of the equation.

$x = 4$

Step 12

Set $x + 8$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 12.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 13

The final solution is all the values that make $(x - 4) (x + 8) = 0$ true.

$x = 4, - 8$

Step 14

Exclude the solutions that do not make $2 \log_{4} (x + 4) - \log_{4} (x + 12) = 1$ true.

$x = 4$