Precalculus Examples

$\ln (3 + x) + \ln (2 x - 1) - 2 \ln (x)$

Step 1

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

$\ln ((3 + x) (2 x - 1)) - 2 \ln (x)$

Step 2

Simplify each term.

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Step 2.1

Expand $(3 + x) (2 x - 1)$ using the FOIL Method.

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Step 2.1.1

Apply the distributive property.

$\ln (3 (2 x - 1) + x (2 x - 1)) - 2 \ln (x)$

Step 2.1.2

Apply the distributive property.

$\ln (3 (2 x) + 3 \cdot - 1 + x (2 x - 1)) - 2 \ln (x)$

Step 2.1.3

Apply the distributive property.

$\ln (3 (2 x) + 3 \cdot - 1 + x (2 x) + x \cdot - 1) - 2 \ln (x)$

Step 2.2

Simplify and combine like terms.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $2$ by $3$ .

$\ln (6 x + 3 \cdot - 1 + x (2 x) + x \cdot - 1) - 2 \ln (x)$

Step 2.2.1.2

Multiply $3$ by $- 1$ .

$\ln (6 x - 3 + x (2 x) + x \cdot - 1) - 2 \ln (x)$

Step 2.2.1.3

Rewrite using the commutative property of multiplication.

$\ln (6 x - 3 + 2 x \cdot x + x \cdot - 1) - 2 \ln (x)$

Step 2.2.1.4

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

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Step 2.2.1.4.1

Move $x$ .

$\ln (6 x - 3 + 2 (x \cdot x) + x \cdot - 1) - 2 \ln (x)$

Step 2.2.1.4.2

Multiply $x$ by $x$ .

$\ln (6 x - 3 + 2 x^{2} + x \cdot - 1) - 2 \ln (x)$

Step 2.2.1.5

Move $- 1$ to the left of $x$ .

$\ln (6 x - 3 + 2 x^{2} - 1 \cdot x) - 2 \ln (x)$

Step 2.2.1.6

Rewrite $- 1 x$ as $- x$ .

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

Step 2.2.2

Subtract $x$ from $6 x$ .

$\ln (5 x - 3 + 2 x^{2}) - 2 \ln (x)$

Step 2.3

Simplify $- 2 \ln (x)$ by moving $2$ inside the logarithm.

$\ln (5 x - 3 + 2 x^{2}) - \ln (x^{2})$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{5 x - 3 + 2 x^{2}}{x^{2}})$

Step 4

Factor by grouping.

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Step 4.1

Reorder terms.

$\ln (\frac{2 x^{2} + 5 x - 3}{x^{2}})$

Step 4.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = 5$ .

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Step 4.2.1

Factor $5$ out of $5 x$ .

$\ln (\frac{2 x^{2} + 5 (x) - 3}{x^{2}})$

Step 4.2.2

Rewrite $5$ as $- 1$ plus $6$

$\ln (\frac{2 x^{2} + (- 1 + 6) x - 3}{x^{2}})$

Step 4.2.3

Apply the distributive property.

$\ln (\frac{2 x^{2} - 1 x + 6 x - 3}{x^{2}})$

Step 4.3

Factor out the greatest common factor from each group.

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Step 4.3.1

Group the first two terms and the last two terms.

$\ln (\frac{(2 x^{2} - 1 x) + 6 x - 3}{x^{2}})$

Step 4.3.2

Factor out the greatest common factor (GCF) from each group.

$\ln (\frac{x (2 x - 1) + 3 (2 x - 1)}{x^{2}})$

Step 4.4

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$\ln (\frac{(2 x - 1) (x + 3)}{x^{2}})$