Precalculus Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Simplify the expression.

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Step 1.3.1

Multiply $x$ by $x$ .

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

Step 1.3.2

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

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

Step 2

Rewrite $\log_{6} (x^{2} - 3 x) = 2$ 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 $b^{y} = x$ .

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

Step 3

Solve for $x$ .

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Step 3.1

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

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

Step 3.2

Raise $6$ to the power of $2$ .

$x^{2} - 3 x = 36$

Step 3.3

Subtract $36$ from both sides of the equation.

$x^{2} - 3 x - 36 = 0$

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 $a = 1$ , $b = - 3$ , and $c = - 36$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot - 36)}}{2 \cdot 1}$

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 36}}{2 \cdot 1}$

Step 3.6.1.2

Multiply $- 4 \cdot 1 \cdot - 36$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 36}}{2 \cdot 1}$

Step 3.6.1.2.2

Multiply $- 4$ by $- 36$ .

$x = \frac{3 \pm \sqrt{9 + 144}}{2 \cdot 1}$

Step 3.6.1.3

Add $9$ and $144$ .

$x = \frac{3 \pm \sqrt{153}}{2 \cdot 1}$

Step 3.6.1.4

Rewrite $153$ as $3^{2} \cdot 17$ .

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Step 3.6.1.4.1

Factor $9$ out of $153$ .

$x = \frac{3 \pm \sqrt{9 (17)}}{2 \cdot 1}$

Step 3.6.1.4.2

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

$x = \frac{3 \pm \sqrt{3^{2} \cdot 17}}{2 \cdot 1}$

Step 3.6.1.5

Pull terms out from under the radical.

$x = \frac{3 \pm 3 \sqrt{17}}{2 \cdot 1}$

Step 3.6.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm 3 \sqrt{17}}{2}$

Step 3.7

The final answer is the combination of both solutions.

$x = \frac{3 + 3 \sqrt{17}}{2}, \frac{3 - 3 \sqrt{17}}{2}$

Step 4

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

$x = \frac{3 + 3 \sqrt{17}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + 3 \sqrt{17}}{2}$

Decimal Form:

$x = 7.68465843 \dots$