Trigonometry Examples

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

Step 1

Simplify $\ln (1 - x) + \ln (9 - x)$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

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.

Tap for more steps...

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

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = - 1$

Step 3