Trigonometry Examples

$\frac{\log (16 - x^{2})}{\log (3 x - 4)} = 2$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$\log (3 x - 4), 1$

Step 1.2

The LCM of one and any expression is the expression.

$\log (3 x - 4)$

Step 2

Multiply each term in $\frac{\log (16 - x^{2})}{\log (3 x - 4)} = 2$ by $\log (3 x - 4)$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{\log (16 - x^{2})}{\log (3 x - 4)} = 2$ by $\log (3 x - 4)$ .

$\frac{\log (16 - x^{2})}{\log (3 x - 4)} \log (3 x - 4) = 2 \log (3 x - 4)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $\log (3 x - 4)$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{\log (16 - x^{2})}{\log (3 x - 4)} \log (3 x - 4) = 2 \log (3 x - 4)$

Step 2.2.1.2

Rewrite the expression.

$\log (16 - x^{2}) = 2 \log (3 x - 4)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify $2 \log (3 x - 4)$ by moving $2$ inside the logarithm.

$\log (16 - x^{2}) = \log ({(3 x - 4)}^{2})$

Step 3

Solve the equation.

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Step 3.1

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$16 - x^{2} = {(3 x - 4)}^{2}$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Simplify ${(3 x - 4)}^{2}$ .

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Step 3.2.1.1

Rewrite.

$16 - x^{2} = 0 + 0 + {(3 x - 4)}^{2}$

Step 3.2.1.2

Rewrite ${(3 x - 4)}^{2}$ as $(3 x - 4) (3 x - 4)$ .

$16 - x^{2} = (3 x - 4) (3 x - 4)$

Step 3.2.1.3

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

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Step 3.2.1.3.1

Apply the distributive property.

$16 - x^{2} = 3 x (3 x - 4) - 4 (3 x - 4)$

Step 3.2.1.3.2

Apply the distributive property.

$16 - x^{2} = 3 x (3 x) + 3 x \cdot - 4 - 4 (3 x - 4)$

Step 3.2.1.3.3

Apply the distributive property.

$16 - x^{2} = 3 x (3 x) + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4$

Step 3.2.1.4

Simplify and combine like terms.

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Step 3.2.1.4.1

Simplify each term.

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Step 3.2.1.4.1.1

Rewrite using the commutative property of multiplication.

$16 - x^{2} = 3 \cdot 3 x \cdot x + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4$

Step 3.2.1.4.1.2

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

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Step 3.2.1.4.1.2.1

Move $x$ .

$16 - x^{2} = 3 \cdot 3 (x \cdot x) + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4$

Step 3.2.1.4.1.2.2

Multiply $x$ by $x$ .

$16 - x^{2} = 3 \cdot 3 x^{2} + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4$

Step 3.2.1.4.1.3

Multiply $3$ by $3$ .

$16 - x^{2} = 9 x^{2} + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4$

Step 3.2.1.4.1.4

Multiply $- 4$ by $3$ .

$16 - x^{2} = 9 x^{2} - 12 x - 4 (3 x) - 4 \cdot - 4$

Step 3.2.1.4.1.5

Multiply $3$ by $- 4$ .

$16 - x^{2} = 9 x^{2} - 12 x - 12 x - 4 \cdot - 4$

Step 3.2.1.4.1.6

Multiply $- 4$ by $- 4$ .

$16 - x^{2} = 9 x^{2} - 12 x - 12 x + 16$

Step 3.2.1.4.2

Subtract $12 x$ from $- 12 x$ .

$16 - x^{2} = 9 x^{2} - 24 x + 16$

Step 3.2.2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$9 x^{2} - 24 x + 16 = 16 - x^{2}$

Step 3.2.3

Move all terms containing $x$ to the left side of the equation.

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Step 3.2.3.1

Add $x^{2}$ to both sides of the equation.

$9 x^{2} - 24 x + 16 + x^{2} = 16$

Step 3.2.3.2

Add $9 x^{2}$ and $x^{2}$ .

$10 x^{2} - 24 x + 16 = 16$

Step 3.2.4

Subtract $16$ from both sides of the equation.

$10 x^{2} - 24 x + 16 - 16 = 0$

Step 3.2.5

Combine the opposite terms in $10 x^{2} - 24 x + 16 - 16$ .

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Step 3.2.5.1

Subtract $16$ from $16$ .

$10 x^{2} - 24 x + 0 = 0$

Step 3.2.5.2

Add $10 x^{2} - 24 x$ and $0$ .

$10 x^{2} - 24 x = 0$

Step 3.2.6

Factor $2 x$ out of $10 x^{2} - 24 x$ .

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Step 3.2.6.1

Factor $2 x$ out of $10 x^{2}$ .

$2 x (5 x) - 24 x = 0$

Step 3.2.6.2

Factor $2 x$ out of $- 24 x$ .

$2 x (5 x) + 2 x (- 12) = 0$

Step 3.2.6.3

Factor $2 x$ out of $2 x (5 x) + 2 x (- 12)$ .

$2 x (5 x - 12) = 0$

Step 3.2.7

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$5 x - 12 = 0$

Step 3.2.8

Set $x$ equal to $0$ .

$x = 0$

Step 3.2.9

Set $5 x - 12$ equal to $0$ and solve for $x$ .

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Step 3.2.9.1

Set $5 x - 12$ equal to $0$ .

$5 x - 12 = 0$

Step 3.2.9.2

Solve $5 x - 12 = 0$ for $x$ .

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Step 3.2.9.2.1

Add $12$ to both sides of the equation.

$5 x = 12$

Step 3.2.9.2.2

Divide each term in $5 x = 12$ by $5$ and simplify.

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Step 3.2.9.2.2.1

Divide each term in $5 x = 12$ by $5$ .

$\frac{5 x}{5} = \frac{12}{5}$

Step 3.2.9.2.2.2

Simplify the left side.

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Step 3.2.9.2.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.9.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{12}{5}$

Step 3.2.9.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{12}{5}$

Step 3.2.10

The final solution is all the values that make $2 x (5 x - 12) = 0$ true.

$x = 0, \frac{12}{5}$

Step 4

Exclude the solutions that do not make $\frac{\log (16 - x^{2})}{\log (3 x - 4)} = 2$ true.

$x = \frac{12}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{12}{5}$

Decimal Form:

$x = 2.4$

Mixed Number Form:

$x = 2 \frac{2}{5}$