Trigonometry Examples

$\log_{6} (2 x - 1) + \log_{6} (4 x + 3) = 2$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{6} ((2 x - 1) (4 x + 3)) = 2$

Step 1.2

Expand $(2 x - 1) (4 x + 3)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\log_{6} (2 x (4 x + 3) - 1 (4 x + 3)) = 2$

Step 1.2.2

Apply the distributive property.

$\log_{6} (2 x (4 x) + 2 x \cdot 3 - 1 (4 x + 3)) = 2$

Step 1.2.3

Apply the distributive property.

$\log_{6} (2 x (4 x) + 2 x \cdot 3 - 1 (4 x) - 1 \cdot 3) = 2$

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

Rewrite using the commutative property of multiplication.

$\log_{6} (2 \cdot (4 x \cdot x) + 2 x \cdot 3 - 1 (4 x) - 1 \cdot 3) = 2$

Step 1.3.1.2

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

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Step 1.3.1.2.1

Move $x$ .

$\log_{6} (2 \cdot (4 (x \cdot x)) + 2 x \cdot 3 - 1 (4 x) - 1 \cdot 3) = 2$

Step 1.3.1.2.2

Multiply $x$ by $x$ .

$\log_{6} (2 \cdot (4 x^{2}) + 2 x \cdot 3 - 1 (4 x) - 1 \cdot 3) = 2$

Step 1.3.1.3

Multiply $2$ by $4$ .

$\log_{6} (8 x^{2} + 2 x \cdot 3 - 1 (4 x) - 1 \cdot 3) = 2$

Step 1.3.1.4

Multiply $3$ by $2$ .

$\log_{6} (8 x^{2} + 6 x - 1 (4 x) - 1 \cdot 3) = 2$

Step 1.3.1.5

Multiply $4$ by $- 1$ .

$\log_{6} (8 x^{2} + 6 x - 4 x - 1 \cdot 3) = 2$

Step 1.3.1.6

Multiply $- 1$ by $3$ .

$\log_{6} (8 x^{2} + 6 x - 4 x - 3) = 2$

Step 1.3.2

Subtract $4 x$ from $6 x$ .

$\log_{6} (8 x^{2} + 2 x - 3) = 2$

Step 2

Rewrite $\log_{6} (8 x^{2} + 2 x - 3) = 2$ 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 $b^{y} = x$ .

$6^{2} = 8 x^{2} + 2 x - 3$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $8 x^{2} + 2 x - 3 = 6^{2}$ .

$8 x^{2} + 2 x - 3 = 6^{2}$

Step 3.2

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

$8 x^{2} + 2 x - 3 = 36$

Step 3.3

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

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Step 3.3.1

Subtract $36$ from both sides of the equation.

$8 x^{2} + 2 x - 3 - 36 = 0$

Step 3.3.2

Subtract $36$ from $- 3$ .

$8 x^{2} + 2 x - 39 = 0$

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 $a = 8$ , $b = 2$ , and $c = - 39$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (8 \cdot - 39)}}{2 \cdot 8}$

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 $2$ to the power of $2$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 8 \cdot - 39}}{2 \cdot 8}$

Step 3.6.1.2

Multiply $- 4 \cdot 8 \cdot - 39$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{- 2 \pm \sqrt{4 - 32 \cdot - 39}}{2 \cdot 8}$

Step 3.6.1.2.2

Multiply $- 32$ by $- 39$ .

$x = \frac{- 2 \pm \sqrt{4 + 1248}}{2 \cdot 8}$

Step 3.6.1.3

Add $4$ and $1248$ .

$x = \frac{- 2 \pm \sqrt{1252}}{2 \cdot 8}$

Step 3.6.1.4

Rewrite $1252$ as $2^{2} \cdot 313$ .

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Step 3.6.1.4.1

Factor $4$ out of $1252$ .

$x = \frac{- 2 \pm \sqrt{4 (313)}}{2 \cdot 8}$

Step 3.6.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 313}}{2 \cdot 8}$

Step 3.6.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{313}}{2 \cdot 8}$

Step 3.6.2

Multiply $2$ by $8$ .

$x = \frac{- 2 \pm 2 \sqrt{313}}{16}$

Step 3.6.3

Simplify $\frac{- 2 \pm 2 \sqrt{313}}{16}$ .

$x = \frac{- 1 \pm \sqrt{313}}{8}$

Step 3.7

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{313}}{8}, - \frac{1 + \sqrt{313}}{8}$

Step 4

Exclude the solutions that do not make $\log_{6} (2 x - 1) + \log_{6} (4 x + 3) = 2$ true.

$x = - \frac{1 - \sqrt{313}}{8}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 - \sqrt{313}}{8}$

Decimal Form:

$x = 2.08647575 \dots$