Precalculus Examples

$\log (5 x + 1) + 1 + \log (4 x - 7)$

Step 1

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

$\log ((5 x + 1) (4 x - 7)) + 1$

Step 2

Simplify each term.

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Step 2.1

Expand $(5 x + 1) (4 x - 7)$ using the FOIL Method.

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Step 2.1.1

Apply the distributive property.

$\log (5 x (4 x - 7) + 1 (4 x - 7)) + 1$

Step 2.1.2

Apply the distributive property.

$\log (5 x (4 x) + 5 x \cdot - 7 + 1 (4 x - 7)) + 1$

Step 2.1.3

Apply the distributive property.

$\log (5 x (4 x) + 5 x \cdot - 7 + 1 (4 x) + 1 \cdot - 7) + 1$

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

Rewrite using the commutative property of multiplication.

$\log (5 \cdot 4 x \cdot x + 5 x \cdot - 7 + 1 (4 x) + 1 \cdot - 7) + 1$

Step 2.2.1.2

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

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Step 2.2.1.2.1

Move $x$ .

$\log (5 \cdot 4 (x \cdot x) + 5 x \cdot - 7 + 1 (4 x) + 1 \cdot - 7) + 1$

Step 2.2.1.2.2

Multiply $x$ by $x$ .

$\log (5 \cdot 4 x^{2} + 5 x \cdot - 7 + 1 (4 x) + 1 \cdot - 7) + 1$

Step 2.2.1.3

Multiply $5$ by $4$ .

$\log (20 x^{2} + 5 x \cdot - 7 + 1 (4 x) + 1 \cdot - 7) + 1$

Step 2.2.1.4

Multiply $- 7$ by $5$ .

$\log (20 x^{2} - 35 x + 1 (4 x) + 1 \cdot - 7) + 1$

Step 2.2.1.5

Multiply $4 x$ by $1$ .

$\log (20 x^{2} - 35 x + 4 x + 1 \cdot - 7) + 1$

Step 2.2.1.6

Multiply $- 7$ by $1$ .

$\log (20 x^{2} - 35 x + 4 x - 7) + 1$

Step 2.2.2

Add $- 35 x$ and $4 x$ .

$\log (20 x^{2} - 31 x - 7) + 1$