Precalculus Examples

$\log (2 x + 1) + 1 + \log (3 x - 8)$

Step 1

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

$\log ((2 x + 1) (3 x - 8)) + 1$

Step 2

Simplify each term.

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Step 2.1

Expand $(2 x + 1) (3 x - 8)$ using the FOIL Method.

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Step 2.1.1

Apply the distributive property.

$\log (2 x (3 x - 8) + 1 (3 x - 8)) + 1$

Step 2.1.2

Apply the distributive property.

$\log (2 x (3 x) + 2 x \cdot - 8 + 1 (3 x - 8)) + 1$

Step 2.1.3

Apply the distributive property.

$\log (2 x (3 x) + 2 x \cdot - 8 + 1 (3 x) + 1 \cdot - 8) + 1$

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.

$\log (2 \cdot 3 x \cdot x + 2 x \cdot - 8 + 1 (3 x) + 1 \cdot - 8) + 1$

Step 2.2.1.2

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

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Step 2.2.1.2.1

Move $x$ .

$\log (2 \cdot 3 (x \cdot x) + 2 x \cdot - 8 + 1 (3 x) + 1 \cdot - 8) + 1$

Step 2.2.1.2.2

Multiply $x$ by $x$ .

$\log (2 \cdot 3 x^{2} + 2 x \cdot - 8 + 1 (3 x) + 1 \cdot - 8) + 1$

Step 2.2.1.3

Multiply $2$ by $3$ .

$\log (6 x^{2} + 2 x \cdot - 8 + 1 (3 x) + 1 \cdot - 8) + 1$

Step 2.2.1.4

Multiply $- 8$ by $2$ .

$\log (6 x^{2} - 16 x + 1 (3 x) + 1 \cdot - 8) + 1$

Step 2.2.1.5

Multiply $3 x$ by $1$ .

$\log (6 x^{2} - 16 x + 3 x + 1 \cdot - 8) + 1$

Step 2.2.1.6

Multiply $- 8$ by $1$ .

$\log (6 x^{2} - 16 x + 3 x - 8) + 1$

Step 2.2.2

Add $- 16 x$ and $3 x$ .

$\log (6 x^{2} - 13 x - 8) + 1$