Trigonometry Examples

$\ln (3 x - 8) = \ln (2 x + 2) + \ln (x - 2)$

Step 1

Simplify $\ln (2 x + 2) + \ln (x - 2)$ .

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Step 1.1

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

$\ln (3 x - 8) = \ln ((2 x + 2) (x - 2))$

Step 1.2

Expand $(2 x + 2) (x - 2)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\ln (3 x - 8) = \ln (2 x (x - 2) + 2 (x - 2))$

Step 1.2.2

Apply the distributive property.

$\ln (3 x - 8) = \ln (2 x \cdot x + 2 x \cdot - 2 + 2 (x - 2))$

Step 1.2.3

Apply the distributive property.

$\ln (3 x - 8) = \ln (2 x \cdot x + 2 x \cdot - 2 + 2 x + 2 \cdot - 2)$

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 $x$ by adding the exponents.

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Step 1.3.1.1.1

Move $x$ .

$\ln (3 x - 8) = \ln (2 (x \cdot x) + 2 x \cdot - 2 + 2 x + 2 \cdot - 2)$

Step 1.3.1.1.2

Multiply $x$ by $x$ .

$\ln (3 x - 8) = \ln (2 x^{2} + 2 x \cdot - 2 + 2 x + 2 \cdot - 2)$

Step 1.3.1.2

Multiply $- 2$ by $2$ .

$\ln (3 x - 8) = \ln (2 x^{2} - 4 x + 2 x + 2 \cdot - 2)$

Step 1.3.1.3

Multiply $2$ by $- 2$ .

$\ln (3 x - 8) = \ln (2 x^{2} - 4 x + 2 x - 4)$

Step 1.3.2

Add $- 4 x$ and $2 x$ .

$\ln (3 x - 8) = \ln (2 x^{2} - 2 x - 4)$

Step 2

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

No solution

Step 3