Trigonometry Examples

$\log_{4} (x + 7) - \log_{4} (x - 7) = 3$

Step 1

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

$\log_{4} (\frac{x + 7}{x - 7}) = 3$

Step 2

Rewrite $\log_{4} (\frac{x + 7}{x - 7}) = 3$ 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 $b^{y} = x$ .

$4^{3} = \frac{x + 7}{x - 7}$

Step 3

Cross multiply to remove the fraction.

$x + 7 = 4^{3} (x - 7)$

Step 4

Simplify $4^{3} (x - 7)$ .

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Step 4.1

Raise $4$ to the power of $3$ .

$x + 7 = 64 (x - 7)$

Step 4.2

Apply the distributive property.

$x + 7 = 64 x + 64 \cdot - 7$

Step 4.3

Multiply $64$ by $- 7$ .

$x + 7 = 64 x - 448$

Step 5

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

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Step 5.1

Subtract $64 x$ from both sides of the equation.

$x + 7 - 64 x = - 448$

Step 5.2

Subtract $64 x$ from $x$ .

$- 63 x + 7 = - 448$

Step 6

Factor $7$ out of $- 63 x + 7$ .

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Step 6.1

Factor $7$ out of $- 63 x$ .

$7 (- 9 x) + 7 = - 448$

Step 6.2

Factor $7$ out of $7$ .

$7 (- 9 x) + 7 (1) = - 448$

Step 6.3

Factor $7$ out of $7 (- 9 x) + 7 (1)$ .

$7 (- 9 x + 1) = - 448$

Step 7

Divide each term in $7 (- 9 x + 1) = - 448$ by $7$ and simplify.

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Step 7.1

Divide each term in $7 (- 9 x + 1) = - 448$ by $7$ .

$\frac{7 (- 9 x + 1)}{7} = \frac{- 448}{7}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $7$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{7 (- 9 x + 1)}{7} = \frac{- 448}{7}$

Step 7.2.1.2

Divide $- 9 x + 1$ by $1$ .

$- 9 x + 1 = \frac{- 448}{7}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $- 448$ by $7$ .

$- 9 x + 1 = - 64$

Step 8

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

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Step 8.1

Subtract $1$ from both sides of the equation.

$- 9 x = - 64 - 1$

Step 8.2

Subtract $1$ from $- 64$ .

$- 9 x = - 65$

Step 9

Divide each term in $- 9 x = - 65$ by $- 9$ and simplify.

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Step 9.1

Divide each term in $- 9 x = - 65$ by $- 9$ .

$\frac{- 9 x}{- 9} = \frac{- 65}{- 9}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $- 9$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{- 9 x}{- 9} = \frac{- 65}{- 9}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 65}{- 9}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Dividing two negative values results in a positive value.

$x = \frac{65}{9}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{65}{9}$

Decimal Form:

$x = 7.\overline{2}$

Mixed Number Form:

$x = 7 \frac{2}{9}$