Trigonometry Examples

$\log_{4} (x) + \log_{4} (x - 3) = \log_{4} (- 7 x + 21)$

Step 1

Subtract $\log_{4} (- 7 x + 21)$ from both sides of the equation.

$\log_{4} (x) + \log_{4} (x - 3) - \log_{4} (- 7 x + 21) = 0$

Step 2

Simplify $\log_{4} (x) + \log_{4} (x - 3) - \log_{4} (- 7 x + 21)$ .

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Step 2.1

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

$\log_{4} (x (x - 3)) - \log_{4} (- 7 x + 21) = 0$

Step 2.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{4} (\frac{x (x - 3)}{- 7 x + 21}) = 0$

Step 2.3

Factor $7$ out of $- 7 x + 21$ .

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Step 2.3.1

Factor $7$ out of $- 7 x$ .

$\log_{4} (\frac{x (x - 3)}{7 (- x) + 21}) = 0$

Step 2.3.2

Factor $7$ out of $21$ .

$\log_{4} (\frac{x (x - 3)}{7 (- x) + 7 (3)}) = 0$

Step 2.3.3

Factor $7$ out of $7 (- x) + 7 (3)$ .

$\log_{4} (\frac{x (x - 3)}{7 (- x + 3)}) = 0$

Step 2.4

Cancel the common factor of $x - 3$ and $- x + 3$ .

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Step 2.4.1

Factor $- 1$ out of $x$ .

$\log_{4} (\frac{x (- 1 (- x) - 3)}{7 (- x + 3)}) = 0$

Step 2.4.2

Rewrite $- 3$ as $- 1 (3)$ .

$\log_{4} (\frac{x (- 1 (- x) - 1 (3))}{7 (- x + 3)}) = 0$

Step 2.4.3

Factor $- 1$ out of $- 1 (- x) - 1 (3)$ .

$\log_{4} (\frac{x (- 1 (- x + 3))}{7 (- x + 3)}) = 0$

Step 2.4.4

Cancel the common factor.

$\log_{4} (\frac{x (- 1 (- x + 3))}{7 (- x + 3)}) = 0$

Step 2.4.5

Rewrite the expression.

$\log_{4} (\frac{x \cdot (- 1)}{7}) = 0$

Step 2.5

Move $- 1$ to the left of $x$ .

$\log_{4} (\frac{- 1 \cdot x}{7}) = 0$

Step 2.6

Move the negative in front of the fraction.

$\log_{4} (- \frac{x}{7}) = 0$

Step 3

Set the argument in $\log_{4} (- \frac{x}{7})$ less than or equal to $0$ to find where the expression is undefined.

$- \frac{x}{7} \leq 0$

Step 4

Solve for $x$ .

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Step 4.1

Divide each term in $- \frac{x}{7} \leq 0$ by $- 1$ and simplify.

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Step 4.1.1

Divide each term in $- \frac{x}{7} \leq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{7}}{- 1} \geq \frac{0}{- 1}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x}{7}}{1} \geq \frac{0}{- 1}$

Step 4.1.2.2

Divide $\frac{x}{7}$ by $1$ .

$\frac{x}{7} \geq \frac{0}{- 1}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $- 1$ .

$\frac{x}{7} \geq 0$

Step 4.2

Multiply both sides by $7$ .

$\frac{x}{7} \cdot 7 \geq 0 \cdot 7$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

Cancel the common factor of $7$ .

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Step 4.3.1.1.1

Cancel the common factor.

$\frac{x}{7} \cdot 7 \geq 0 \cdot 7$

Step 4.3.1.1.2

Rewrite the expression.

$x \geq 0 \cdot 7$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Multiply $0$ by $7$ .

$x \geq 0$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \geq 0$

$[0, \infty)$

Step 6