Algebra Examples

$\ln (- 2 x + 5) + \ln (x + 4) = k$

Step 1

Rewrite the equation as $k = \ln (- 2 x + 5) + \ln (x + 4)$ .

$k = \ln (- 2 x + 5) + \ln (x + 4)$

Step 2

Simplify $\ln (- 2 x + 5) + \ln (x + 4)$ .

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Step 2.1

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

$k = \ln ((- 2 x + 5) (x + 4))$

Step 2.2

Expand $(- 2 x + 5) (x + 4)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$k = \ln (- 2 x (x + 4) + 5 (x + 4))$

Step 2.2.2

Apply the distributive property.

$k = \ln (- 2 x \cdot x - 2 x \cdot 4 + 5 (x + 4))$

Step 2.2.3

Apply the distributive property.

$k = \ln (- 2 x \cdot x - 2 x \cdot 4 + 5 x + 5 \cdot 4)$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

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

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Step 2.3.1.1.1

Move $x$ .

$k = \ln (- 2 (x \cdot x) - 2 x \cdot 4 + 5 x + 5 \cdot 4)$

Step 2.3.1.1.2

Multiply $x$ by $x$ .

$k = \ln (- 2 x^{2} - 2 x \cdot 4 + 5 x + 5 \cdot 4)$

Step 2.3.1.2

Multiply $4$ by $- 2$ .

$k = \ln (- 2 x^{2} - 8 x + 5 x + 5 \cdot 4)$

Step 2.3.1.3

Multiply $5$ by $4$ .

$k = \ln (- 2 x^{2} - 8 x + 5 x + 20)$

Step 2.3.2

Add $- 8 x$ and $5 x$ .

$k = \ln (- 2 x^{2} - 3 x + 20)$