Precalculus Examples

$\log_{9} (x + 6) + \log_{9} (x + 3) = 2$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{9} ((x + 6) (x + 3)) = 2$

Step 1.2

Expand $(x + 6) (x + 3)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\log_{9} (x (x + 3) + 6 (x + 3)) = 2$

Step 1.2.2

Apply the distributive property.

$\log_{9} (x \cdot x + x \cdot 3 + 6 (x + 3)) = 2$

Step 1.2.3

Apply the distributive property.

$\log_{9} (x \cdot x + x \cdot 3 + 6 x + 6 \cdot 3) = 2$

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

Multiply $x$ by $x$ .

$\log_{9} (x^{2} + x \cdot 3 + 6 x + 6 \cdot 3) = 2$

Step 1.3.1.2

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

$\log_{9} (x^{2} + 3 \cdot x + 6 x + 6 \cdot 3) = 2$

Step 1.3.1.3

Multiply $6$ by $3$ .

$\log_{9} (x^{2} + 3 x + 6 x + 18) = 2$

Step 1.3.2

Add $3 x$ and $6 x$ .

$\log_{9} (x^{2} + 9 x + 18) = 2$

Step 2

Rewrite $\log_{9} (x^{2} + 9 x + 18) = 2$ 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 $b^{y} = x$ .

$9^{2} = x^{2} + 9 x + 18$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $x^{2} + 9 x + 18 = 9^{2}$ .

$x^{2} + 9 x + 18 = 9^{2}$

Step 3.2

Raise $9$ to the power of $2$ .

$x^{2} + 9 x + 18 = 81$

Step 3.3

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

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Step 3.3.1

Subtract $81$ from both sides of the equation.

$x^{2} + 9 x + 18 - 81 = 0$

Step 3.3.2

Subtract $81$ from $18$ .

$x^{2} + 9 x - 63 = 0$

Step 3.4

Use the quadratic formula to find the solutions.

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

Step 3.5

Substitute the values $a = 1$ , $b = 9$ , and $c = - 63$ into the quadratic formula and solve for $x$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (1 \cdot - 63)}}{2 \cdot 1}$

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 1 \cdot - 63}}{2 \cdot 1}$

Step 3.6.1.2

Multiply $- 4 \cdot 1 \cdot - 63$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot - 63}}{2 \cdot 1}$

Step 3.6.1.2.2

Multiply $- 4$ by $- 63$ .

$x = \frac{- 9 \pm \sqrt{81 + 252}}{2 \cdot 1}$

Step 3.6.1.3

Add $81$ and $252$ .

$x = \frac{- 9 \pm \sqrt{333}}{2 \cdot 1}$

Step 3.6.1.4

Rewrite $333$ as $3^{2} \cdot 37$ .

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Step 3.6.1.4.1

Factor $9$ out of $333$ .

$x = \frac{- 9 \pm \sqrt{9 (37)}}{2 \cdot 1}$

Step 3.6.1.4.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{- 9 \pm \sqrt{3^{2} \cdot 37}}{2 \cdot 1}$

Step 3.6.1.5

Pull terms out from under the radical.

$x = \frac{- 9 \pm 3 \sqrt{37}}{2 \cdot 1}$

Step 3.6.2

Multiply $2$ by $1$ .

$x = \frac{- 9 \pm 3 \sqrt{37}}{2}$

Step 3.7

The final answer is the combination of both solutions.

$x = - \frac{9 - 3 \sqrt{37}}{2}, - \frac{9 + 3 \sqrt{37}}{2}$

Step 4

Exclude the solutions that do not make $\log_{9} (x + 6) + \log_{9} (x + 3) = 2$ true.

$x = - \frac{9 - 3 \sqrt{37}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{9 - 3 \sqrt{37}}{2}$

Decimal Form:

$x = 4.62414379 \dots$