Trigonometry Examples

$\ln (1 - x) + \ln (9 - x) = \ln (20)$

Step 1

Simplify the left side.

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Step 1.1

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

$\ln ((1 - x) (9 - x)) = \ln (20)$

Step 1.2

Expand $(1 - x) (9 - x)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\ln (1 (9 - x) - x (9 - x)) = \ln (20)$

Step 1.2.2

Apply the distributive property.

$\ln (1 \cdot 9 + 1 (- x) - x (9 - x)) = \ln (20)$

Step 1.2.3

Apply the distributive property.

$\ln (1 \cdot 9 + 1 (- x) - x \cdot 9 - x (- x)) = \ln (20)$

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 $9$ by $1$ .

$\ln (9 + 1 (- x) - x \cdot 9 - x (- x)) = \ln (20)$

Step 1.3.1.2

Multiply $- x$ by $1$ .

$\ln (9 - x - x \cdot 9 - x (- x)) = \ln (20)$

Step 1.3.1.3

Multiply $9$ by $- 1$ .

$\ln (9 - x - 9 x - x (- x)) = \ln (20)$

Step 1.3.1.4

Rewrite using the commutative property of multiplication.

$\ln (9 - x - 9 x - 1 \cdot (- 1 x \cdot x)) = \ln (20)$

Step 1.3.1.5

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

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Step 1.3.1.5.1

Move $x$ .

$\ln (9 - x - 9 x - 1 \cdot (- 1 (x \cdot x))) = \ln (20)$

Step 1.3.1.5.2

Multiply $x$ by $x$ .

$\ln (9 - x - 9 x - 1 \cdot (- 1 x^{2})) = \ln (20)$

Step 1.3.1.6

Multiply $- 1$ by $- 1$ .

$\ln (9 - x - 9 x + 1 x^{2}) = \ln (20)$

Step 1.3.1.7

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

$\ln (9 - x - 9 x + x^{2}) = \ln (20)$

Step 1.3.2

Subtract $9 x$ from $- x$ .

$\ln (9 - 10 x + x^{2}) = \ln (20)$

Step 2

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

$9 - 10 x + x^{2} = 20$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $20$ from both sides of the equation.

$9 - 10 x + x^{2} - 20 = 0$

Step 3.2

Subtract $20$ from $9$ .

$- 10 x + x^{2} - 11 = 0$

Step 3.3

Factor the left side of the equation.

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Step 3.3.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$- 10 u + u^{2} - 11 = 0$

Step 3.3.2

Factor $- 10 u + u^{2} - 11$ using the AC method.

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Step 3.3.2.1

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 $- 11$ and whose sum is $- 10$ .

$- 11, 1$

Step 3.3.2.2

Write the factored form using these integers.

$(u - 11) (u + 1) = 0$

Step 3.3.3

Replace all occurrences of $u$ with $x$ .

$(x - 11) (x + 1) = 0$

Step 3.4

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

$x - 11 = 0$

$x + 1 = 0$

Step 3.5

Set $x - 11$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x - 11$ equal to $0$ .

$x - 11 = 0$

Step 3.5.2

Add $11$ to both sides of the equation.

$x = 11$

Step 3.6

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 3.6.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 3.7

The final solution is all the values that make $(x - 11) (x + 1) = 0$ true.

$x = 11, - 1$

Step 4

Exclude the solutions that do not make $\ln (1 - x) + \ln (9 - x) = \ln (20)$ true.

$x = - 1$