Trigonometry Examples

$\log_{3} (x - 7) = 2 - \log_{3} (x + 1)$

Step 1

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

$\log_{3} (x - 7) + \log_{3} (x + 1) = 2$

Step 2

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

$\log_{3} ((x - 7) (x + 1)) = 2$

Step 3

Expand $(x - 7) (x + 1)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\log_{3} (x (x + 1) - 7 (x + 1)) = 2$

Step 3.2

Apply the distributive property.

$\log_{3} (x \cdot x + x \cdot 1 - 7 (x + 1)) = 2$

Step 3.3

Apply the distributive property.

$\log_{3} (x \cdot x + x \cdot 1 - 7 x - 7 \cdot 1) = 2$

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 $x$ by $x$ .

$\log_{3} (x^{2} + x \cdot 1 - 7 x - 7 \cdot 1) = 2$

Step 4.1.2

Multiply $x$ by $1$ .

$\log_{3} (x^{2} + x - 7 x - 7 \cdot 1) = 2$

Step 4.1.3

Multiply $- 7$ by $1$ .

$\log_{3} (x^{2} + x - 7 x - 7) = 2$

Step 4.2

Subtract $7 x$ from $x$ .

$\log_{3} (x^{2} - 6 x - 7) = 2$

Step 5

Rewrite $\log_{3} (x^{2} - 6 x - 7) = 2$ 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 $b^{y} = x$ .

$3^{2} = x^{2} - 6 x - 7$

Step 6

Solve for $x$ .

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Step 6.1

Rewrite the equation as $x^{2} - 6 x - 7 = 3^{2}$ .

$x^{2} - 6 x - 7 = 3^{2}$

Step 6.2

Raise $3$ to the power of $2$ .

$x^{2} - 6 x - 7 = 9$

Step 6.3

Subtract $9$ from both sides of the equation.

$x^{2} - 6 x - 7 - 9 = 0$

Step 6.4

Subtract $9$ from $- 7$ .

$x^{2} - 6 x - 16 = 0$

Step 6.5

Factor $x^{2} - 6 x - 16$ using the AC method.

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Step 6.5.1

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 $- 16$ and whose sum is $- 6$ .

$- 8, 2$

Step 6.5.2

Write the factored form using these integers.

$(x - 8) (x + 2) = 0$

Step 6.6

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

$x - 8 = 0$

$x + 2 = 0$

Step 6.7

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

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Step 6.7.1

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

$x - 8 = 0$

Step 6.7.2

Add $8$ to both sides of the equation.

$x = 8$

Step 6.8

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 6.8.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 6.8.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6.9

The final solution is all the values that make $(x - 8) (x + 2) = 0$ true.

$x = 8, - 2$

Step 7

Exclude the solutions that do not make $\log_{3} (x - 7) = 2 - \log_{3} (x + 1)$ true.

$x = 8$