Trigonometry Examples

$\log (3 x) = \log (5) + \log (x - 4)$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\log (5)$ from both sides of the equation.

$\log (3 x) - \log (5) = \log (x - 4)$

Step 1.2

Subtract $\log (x - 4)$ from both sides of the equation.

$\log (3 x) - \log (5) - \log (x - 4) = 0$

Step 2

Simplify $\log (3 x) - \log (5) - \log (x - 4)$ .

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Step 2.1

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

$\log (\frac{3 x}{5}) - \log (x - 4) = 0$

Step 2.2

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

$\log (\frac{\frac{3 x}{5}}{x - 4}) = 0$

Step 2.3

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{3 x}{5} \cdot \frac{1}{x - 4}) = 0$

Step 2.4

Multiply $\frac{3 x}{5}$ by $\frac{1}{x - 4}$ .

$\log (\frac{3 x}{5 (x - 4)}) = 0$

Step 3

Set the denominator in $\frac{3 x}{5 (x - 4)}$ equal to $0$ to find where the expression is undefined.

$5 (x - 4) = 0$

Step 4

Solve for $x$ .

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Step 4.1

Divide each term in $5 (x - 4) = 0$ by $5$ and simplify.

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Step 4.1.1

Divide each term in $5 (x - 4) = 0$ by $5$ .

$\frac{5 (x - 4)}{5} = \frac{0}{5}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $5$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{5 (x - 4)}{5} = \frac{0}{5}$

Step 4.1.2.1.2

Divide $x - 4$ by $1$ .

$x - 4 = \frac{0}{5}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $5$ .

$x - 4 = 0$

Step 4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 5

Set the argument in $\log (\frac{3 x}{5 (x - 4)})$ less than or equal to $0$ to find where the expression is undefined.

$\frac{3 x}{5 (x - 4)} \leq 0$

Step 6

Solve for $x$ .

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Step 6.1

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$3 x = 0$

$x - 4 = 0$

Step 6.2

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 6.2.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $3$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $0$ by $3$ .

$x = 0$

Step 6.3

Add $4$ to both sides of the equation.

$x = 4$

Step 6.4

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = 0$

$x = 4$

Step 6.5

Consolidate the solutions.

$x = 0, 4$

Step 6.6

Find the domain of $\frac{3 x}{5 (x - 4)}$ .

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Step 6.6.1

Set the denominator in $\frac{3 x}{5 (x - 4)}$ equal to $0$ to find where the expression is undefined.

$5 (x - 4) = 0$

Step 6.6.2

Solve for $x$ .

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Step 6.6.2.1

Divide each term in $5 (x - 4) = 0$ by $5$ and simplify.

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Step 6.6.2.1.1

Divide each term in $5 (x - 4) = 0$ by $5$ .

$\frac{5 (x - 4)}{5} = \frac{0}{5}$

Step 6.6.2.1.2

Simplify the left side.

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Step 6.6.2.1.2.1

Cancel the common factor of $5$ .

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Step 6.6.2.1.2.1.1

Cancel the common factor.

$\frac{5 (x - 4)}{5} = \frac{0}{5}$

Step 6.6.2.1.2.1.2

Divide $x - 4$ by $1$ .

$x - 4 = \frac{0}{5}$

Step 6.6.2.1.3

Simplify the right side.

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Step 6.6.2.1.3.1

Divide $0$ by $5$ .

$x - 4 = 0$

Step 6.6.2.2

Add $4$ to both sides of the equation.

$x = 4$

Step 6.6.3

The domain is all values of $x$ that make the expression defined.

$(- \infty, 4) \cup (4, \infty)$

Step 6.7

Use each root to create test intervals.

$x < 0$

$0 < x < 4$

$x > 4$

Step 6.8

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 6.8.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 6.8.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 6.8.1.2

Replace $x$ with $- 2$ in the original inequality.

$\frac{3 (- 2)}{5 ((- 2) - 4)} \leq 0$

Step 6.8.1.3

The left side $0.2$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 6.8.2

Test a value on the interval $0 < x < 4$ to see if it makes the inequality true.

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Step 6.8.2.1

Choose a value on the interval $0 < x < 4$ and see if this value makes the original inequality true.

$x = 2$

Step 6.8.2.2

Replace $x$ with $2$ in the original inequality.

$\frac{3 (2)}{5 ((2) - 4)} \leq 0$

Step 6.8.2.3

The left side $- 0.6$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 6.8.3

Test a value on the interval $x > 4$ to see if it makes the inequality true.

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Step 6.8.3.1

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Step 6.8.3.2

Replace $x$ with $6$ in the original inequality.

$\frac{3 (6)}{5 ((6) - 4)} \leq 0$

Step 6.8.3.3

The left side $1.8$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 6.8.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < 0$ False

$0 < x < 4$ True

$x > 4$ False

$x < 0$ False

$0 < x < 4$ True

$x > 4$ False

Step 6.9

The solution consists of all of the true intervals.

$0 \leq x < 4$

Step 7

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 $0$ .

$0 \leq x \leq 4$

$[0, 4]$

Step 8