Precalculus Examples

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

Step 1

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

$\ln ((x - 2) (2 x - 3))$

Step 2

Expand $(x - 2) (2 x - 3)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\ln (x (2 x - 3) - 2 (2 x - 3))$

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.1.2

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

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Step 3.1.2.1

Move $x$ .

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

Step 3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.1.3

Move $- 3$ to the left of $x$ .

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

Step 3.1.4

Multiply $2$ by $- 2$ .

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

Step 3.1.5

Multiply $- 2$ by $- 3$ .

$\ln (2 x^{2} - 3 x - 4 x + 6)$

Step 3.2

Subtract $4 x$ from $- 3 x$ .

$\ln (2 x^{2} - 7 x + 6)$