Trigonometry Examples

$\ln (x - 6) + \ln (x - 5) = \ln (16 - 2 x)$

Step 1

Simplify the left side.

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Step 1.1

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

$\ln ((x - 6) (x - 5)) = \ln (16 - 2 x)$

Step 1.2

Expand $(x - 6) (x - 5)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\ln (x (x - 5) - 6 (x - 5)) = \ln (16 - 2 x)$

Step 1.2.2

Apply the distributive property.

$\ln (x \cdot x + x \cdot - 5 - 6 (x - 5)) = \ln (16 - 2 x)$

Step 1.2.3

Apply the distributive property.

$\ln (x \cdot x + x \cdot - 5 - 6 x - 6 \cdot - 5) = \ln (16 - 2 x)$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$\ln (x^{2} + x \cdot - 5 - 6 x - 6 \cdot - 5) = \ln (16 - 2 x)$

Step 1.3.1.2

Move $- 5$ to the left of $x$ .

$\ln (x^{2} - 5 \cdot x - 6 x - 6 \cdot - 5) = \ln (16 - 2 x)$

Step 1.3.1.3

Multiply $- 6$ by $- 5$ .

$\ln (x^{2} - 5 x - 6 x + 30) = \ln (16 - 2 x)$

Step 1.3.2

Subtract $6 x$ from $- 5 x$ .

$\ln (x^{2} - 11 x + 30) = \ln (16 - 2 x)$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$x^{2} - 11 x + 30 = 16 - 2 x$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Add $2 x$ to both sides of the equation.

$x^{2} - 11 x + 30 + 2 x = 16$

Step 3.1.2

Add $- 11 x$ and $2 x$ .

$x^{2} - 9 x + 30 = 16$

Step 3.2

Subtract $16$ from both sides of the equation.

$x^{2} - 9 x + 30 - 16 = 0$

Step 3.3

Subtract $16$ from $30$ .

$x^{2} - 9 x + 14 = 0$

Step 3.4

Factor $x^{2} - 9 x + 14$ using the AC method.

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Step 3.4.1

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 $14$ and whose sum is $- 9$ .

$- 7, - 2$

Step 3.4.2

Write the factored form using these integers.

$(x - 7) (x - 2) = 0$

Step 3.5

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

$x - 7 = 0$

$x - 2 = 0$

Step 3.6

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 3.6.2

Add $7$ to both sides of the equation.

$x = 7$

Step 3.7

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

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Step 3.7.1

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

$x - 2 = 0$

Step 3.7.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.8

The final solution is all the values that make $(x - 7) (x - 2) = 0$ true.

$x = 7, 2$

Step 4

Exclude the solutions that do not make $\ln (x - 6) + \ln (x - 5) = \ln (16 - 2 x)$ true.

$x = 7$