Precalculus Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.2

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

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Step 1.3.1.2.1

Move $x$ .

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

Step 1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 1.3.1.3

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

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

Step 1.3.1.4

Multiply $3$ by $- 2$ .

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

Step 1.3.1.5

Multiply $- 2$ by $3$ .

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

Step 1.3.2

Subtract $6 x$ from $3 x$ .

$\log (3 x^{2} - 3 x - 6) = 1$

Step 2

Rewrite $\log (3 x^{2} - 3 x - 6) = 1$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{1} = 3 x^{2} - 3 x - 6$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $3 x^{2} - 3 x - 6 = 10^{1}$ .

$3 x^{2} - 3 x - 6 = 10$

Step 3.2

Move all terms to the left side of the equation and simplify.

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Step 3.2.1

Subtract $10$ from both sides of the equation.

$3 x^{2} - 3 x - 6 - 10 = 0$

Step 3.2.2

Subtract $10$ from $- 6$ .

$3 x^{2} - 3 x - 16 = 0$

Step 3.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.4

Substitute the values $a = 3$ , $b = - 3$ , and $c = - 16$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (3 \cdot - 16)}}{2 \cdot 3}$

Step 3.5

Simplify.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 3 \cdot - 16}}{2 \cdot 3}$

Step 3.5.1.2

Multiply $- 4 \cdot 3 \cdot - 16$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{3 \pm \sqrt{9 - 12 \cdot - 16}}{2 \cdot 3}$

Step 3.5.1.2.2

Multiply $- 12$ by $- 16$ .

$x = \frac{3 \pm \sqrt{9 + 192}}{2 \cdot 3}$

Step 3.5.1.3

Add $9$ and $192$ .

$x = \frac{3 \pm \sqrt{201}}{2 \cdot 3}$

Step 3.5.2

Multiply $2$ by $3$ .

$x = \frac{3 \pm \sqrt{201}}{6}$

Step 3.6

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{201}}{6}, \frac{3 - \sqrt{201}}{6}$

Step 4

Exclude the solutions that do not make $\log (x - 2) + \log (3 x + 3) = 1$ true.

$x = \frac{3 + \sqrt{201}}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{201}}{6}$

Decimal Form:

$x = 2.86290781 \dots$